FYI: If whoever's name the van is in now gifts it to the other person, you can probably sidestep paying the tax on it. My ex and I did it when we changed our registration from FL to NY. (The clerk at the DMV actually suggested it to us when we about fell over after she said it'd be $1100 bucks initially.)
Have you looked at the list of what's checked on inspection in MD? A timing belt is a maintenace item that I can't imagine triggering an inspection violation. I suspect someone might be trying to take you for a ride. If its not overheating it doesn't need a water pump. Typically people just change them when they're doing the timing because you're already paying for the labor and a water pump itself is cheap. If you can get away without spending big $$ on it, I would drive it until the wheels fall off. I would get a second opinion. (Do not take it to the Toyota dealer btw.)
Moving on, she should do the $1 trial at CreditCheckTotal and get her true FICO shores. Credit Karma, etc., is good for keeping an eye on things, but they provide Vantage scores which basically no lenders actually use. You mind find her scores at CCT are better than you think.
Credit cards don't cause DTI issues unless you are carrying balances. She needs a third credit card to optimize her score. If the home purchase is less than 6 months away, they might be sensitive to the hard pulls for the new card. If it's more than 6 months away and INQs are relatively low, I wouldn't so much worry about a carefully considered app for a third CC.
Definitely try your credit unions for financing and see who has the best rate.
So long as your down payment is more than the tax, tag and title you won't be financing it.
Also don't rule out just buying another few grand car for cash. What will be 5 years out of a $3800 minivan is a pretty darn good run 
Want to address you both in this post [Thanks for the awesome advice]:
When we bought the van, it was AS-IS, so we knew it would not be perfect but were not expecting to have to put so much into it. For example, we don't use the dealer mechanic unless like with our corolla, its covered under warranty. We used a myMechanic guy, and he had to replace cv joints and replace an outside fixture which allows you to get to the timing belt which cost us about $1100 [included cost of oil, inspection parts and labor for joint and sparks something (not the spark plug but something to do with the cylinders). That was just to get us up here to MD in the first. Renting a UHAUL was going to cost us $700 plus mileage and we'd still need to make use of the van. So, we bit the bullet and fixed the van. Our mechanic at the time did say we could get another 10k miles before we MUST change the timing belt(water pump had not been or belt had not been changed since the original year of the car which is 2005).
The brakes is really a safety hazard...I had an incident wear the brakes failed...had lost control and it was raining (not talking hydroplaning) I literally could not brake and had to wait for it to stop on its own.Told my wife about, she was skeptical at first but since she is the main driver of the vehicle, I had her test it. She took for a drive in the cul de sac and she slid off the payment, through mud and almost hit a tree. So there is definitely something wrong with the rear brakes and it scares me that my wife and kids had been driving in it with it faulty like that. The MD mechanic didn't even charge for the brake inspection and all the hours spent trying to see the problem was, just for the oil change(about $60 for sythetic)[this guy is from a mom/pop place here near FT meade, so he is an honest guy]. He said for sme reason you back brakes are not working and suspected it to be a sensor. sensors in this van are expensive. There is another sensor we needed to fix too but the FL mechanic said its not detrimental but would cost a mile or two with fuel economy.
Long story short our corolla purrs like a kitty because we were second owners and the first owners was an elderly couple , it only had about 3k miles at the time. We take care of our cars, pamper them. Our 2010 corolla has barely 65k miles on it. So, due to not wanting to pay $700+ just to register it here in MD next Spring, pay $600 for a timing belt, $1500-2k on NEEDED brake job [it snows pretty well and I can't be cheap with my family's safety], it's starting to become a pit. And I would HATE if something else needed fixing after all this.
We hope a new secondary vehicle will be WELL maintained like our corolla and will probably only have 50k miles on it in 7 years too. Which means we can keep them for well beyond 10+ years. I hope for our Sienna to last until after we close, but by March for sure we will need something reliable.
For context:
The income listed was hers(50-58k), she will be working for DOD God willing. No news is good news as she has told me about OPM. I make 75k but she did not want me to go on the auto loan since I will be the only person on the home loan(we plan to buy w/n 6months or so/before our lease ends on March 1st/tired of pay almost $2k in rent). [I had considered] putting her on the home loan but with her student loans it would increase our DTI dramtically and only got us like an extra 5k of borrowing power than what I get now on my own. Conservative number for a home: 200-250k and ABSOLUTE MAX 300k just based on my file, unless we are willing to pay more for a down payment to cover the difference[thinking no/only can realitically save 12k by March for DP and would NEED closing covered].
Tangent:
If she applied for another CC, how long would she need to wait before getting an auto loan? [future plans: she wants either a fiat 500x Trekking or a Rav4(which I'd prefer she get come that time)] But hey happy wife, happy life, huh! 
I with you!:
If wasn't for the time we lost control, would keep it another 3-5 years. Our mechanic said short trips are fine until we decide what we wanted to do, but would not push LONG distance trips. Like driving from here to NY. And yes, that was our original plan to ride it until it couldn't go NO more but this thing called life you know! And As long as it lasts until at least March when I get a bonus from work, it can fall apart on closing day for allI care! xD
@ejm201 wrote:Want to address you both in this post [Thanks for the awesome advice]:
When we bought the van, it was AS-IS, so we knew it would not be perfect but were not expecting to have to put so much into it. For example, we don't use the dealer mechanic unless like with our corolla, its covered under warranty. We used a myMechanic guy, and he had to replace cv joints and replace an outside fixture which allows you to get to the timing belt which cost us about $1100 [included cost of oil, inspection parts and labor for joint and sparks something (not the spark plug but something to do with the cylinders). That was just to get us up here to MD in the first. Renting a UHAUL was going to cost us $700 plus mileage and we'd still need to make use of the van. So, we bit the bullet and fixed the van. Our mechanic at the time did say we could get another 10k miles before we MUST change the timing belt(water pump had not been or belt had not been changed since the original year of the car which is 2005).
The brakes is really a safety hazard...I had an incident wear the brakes failed...had lost control and it was raining (not talking hydroplaning) I literally could not brake and had to wait for it to stop on its own.Told my wife about, she was skeptical at first but since she is the main driver of the vehicle, I had her test it. She took for a drive in the cul de sac and she slid off the payment, through mud and almost hit a tree. So there is definitely something wrong with the rear brakes and it scares me that my wife and kids had been driving in it with it faulty like that. The MD mechanic didn't even charge for the brake inspection and all the hours spent trying to see the problem was, just for the oil change(about $60 for sythetic)[this guy is from a mom/pop place here near FT meade, so he is an honest guy]. He said for sme reason you back brakes are not working and suspected it to be a sensor. sensors in this van are expensive. There is another sensor we needed to fix too but the FL mechanic said its not detrimental but would cost a mile or two with fuel economy.
Long story short our corolla purrs like a kitty because we were second owners and the first owners was an elderly couple , it only had about 3k miles at the time. We take care of our cars, pamper them. Our 2010 corolla has barely 65k miles on it. So, due to not wanting to pay $700+ just to register it here in MD next Spring, pay $600 for a timing belt, $1500-2k on NEEDED brake job [it snows pretty well and I can't be cheap with my family's safety], it's starting to become a pit. And I would HATE if something else needed fixing after all this.
We hope a new secondary vehicle will be WELL maintained like our corolla and will probably only have 50k miles on it in 7 years too. Which means we can keep them for well beyond 10+ years. I hope for our Sienna to last until after we close, but by March for sure we will need something reliable.
For context:
The income listed was hers(50-58k), she will be working for DOD God willing. No news is good news as she has told me about OPM. I make 75k but she did not want me to go on the auto loan since I will be the only person on the home loan(we plan to buy w/n 6months or so/before our lease ends on March 1st/tired of pay almost $2k in rent). [I had considered] putting her on the home loan but with her student loans it would increase our DTI dramtically and only got us like an extra 5k of borrowing power than what I get now on my own. Conservative number for a home: 200-250k and ABSOLUTE MAX 300k just based on my file, unless we are willing to pay more for a down payment to cover the difference[thinking no/only can realitically save 12k by March for DP and would NEED closing covered].
Tangent:
If she applied for another CC, how long would she need to wait before getting an auto loan? [future plans: she wants either a fiat 500x Trekking or a Rav4(which I'd prefer she get come that time)] But hey happy wife, happy life, huh!
I with you!:
If wasn't for the time we lost control, would keep it another 3-5 years. Our mechanic said short trips are fine until we decide what we wanted to do, but would not push LONG distance trips. Like driving from here to NY. And yes, that was our original plan to ride it until it couldn't go NO more but this thing called life you know! And As long as it lasts until at least March when I get a bonus from work, it can fall apart on closing day for allI care! xD
Got it. Bye Bye Sienna! LOL.
What are the inquiries like on her reports now? Auto loans aren't as sensitive to them as mortgages. You never know, but I don't think one carefully calculated inquiry for a new credit card would hinder the auto loan process.
The CapOne auto prequal is a good idea . . it's a SP so even if it's just for fun it may give you some confidence before you talk to your CU's about a loan.
I think you could get into a low mileage few year old used Rav4 for a good price. And with a similar experience to your Corolla instead of the Sienna!