Paying off car loan 1 a few months early is highly likely to help your mortgage scores a bit, and shouldn't hurt them since you'll still have an open installment tradeline reporting (car loan 2). The mortgage scores really like to see zero balance accounts, so you'll add one by paying off that car loan a little early. You won't suffer the classic "no installment loans" penalty that a lot of folks see when they pay off their only installment tradeline. That account is going to go to zero/paid/closed before you close on a mortgage anyways, so wrapping it up a few months early is not going to be super impactful in any direction. It could help a little, it shouldn't hurt.
We are in a somewhat similar position to you in terms of price range and mortgage scores, our gross household income is a little bit higher. We are not first time buyers and will be rolling equity over out of our current house once we sell it. We had a teleconference with our mortgage broker last week, who told us with our stats/qualifications we should shop without worry and they'd write a prequal letter in that $600k range whenever we need it. He's familiar with our profile and knows us from landing us our current mortgage (and guiding us through refi when rates dropped). Current market conditions in our area sound similar to what you describe (white hot and rising).
In order to boost my middle mortgage score, I am paying off some low balance installment loans and going AZEO on CCs to maximize the zero balance accounts. On a parallel track we are saving every penny to boost our down payment/closing cost cash fund. Once the loans and CCs are paid to zero every extra penny will go into that fund. A 20% down payment is a noble goal if you can get there, but they will likely write you a mortgage all the way to 5% down (or even 3.5% with some products). You'll pay more monthly in rate and PMI, but that's the cost of landing a home these days without spending years accumulating a 20% cash down payment plus cash for closing costs on a $600k house. The PMI comes off in 5-7 years on a standard payment schedule, maybe sooner if the market keeps rising and you get a higher appraisal in a year or two to bridge the equity gap. The higher rate can be refinanced later if a lower one presents itself.
One other thing to keep in mind in your price range is the max conforming limit: $548k. If you have to borrow more than this, you go into jumbo territory. I've heard anecdotally that the jumbo market has tightened considerably in the past year, and you may have trouble qualifying. The rates are also higher. So if you're looking at a $600k property, you need at least $50k down plus your closing costs (3-5% of purchase price, so another $30k cash) in order to stay under that cap and get a conventional conforming loan. Can you raise $80k in cash in the next 3-6 months? If the answer is no, you either need to buy a significantly cheaper house, or forget buying this year and save as much cash as possible until you do have it.
We're essentially doing option 3 from your list, except we're still shopping in the present, and open to structuring a deal with as little as 5% down if our dream house presents itself and we need to act fast. We have the benefit of cashing out the equity in our current house, which will get us almost all the way there. I'd love to put the 20% down for the better rate and no PMI, but it's just not a realistic possibility if we buy in 2021.
