I've never had a "regular" mortgage so I personally can't verify what happens during one. I only have data points from what others have posted over the years. I'd guess that a single credit card isn't a huge issue but make sure to have your reasons planned if they ask about it.
It is standard for new homeowners (who have closed on the loan and have the keys) to go on an app spree the day they close or the day after but not before, so I can't tell you what the risks are specifically.
Discover is a great card but it doesn't work internationally well so don't expect to use it as your backup. I like Capital One for travel because they typically don't charge a foreign transaction fee and they're Visa or Mastercard which is accepted almost everywhere. I go to India once a year on average (typically Mumbai and Goa) and have never had an issue with a Capital One card being accepted everywhere that takes cards.
Most important is to start cutting back luxury spending so you can start saving up cash for the burdens of homeownership. Credit is not enough -- you need to build a strong safety net of savings. I've owned properties for 25+ years and on average I'll have a $3000 surprise cost annually (I do own multiple properties though so it's probably more like $500-$1000 per year per property average). Next year I have to budget for a new roof and driveway ($20,000 total) and this year I had to budget for a new HVAC system ($7000 total) all on the same property. Don't expect to finance everything because the interest charged will be extreme!
Congrats on your pending home purchase, and once you own it and have the keys, look into getting some better cards but don't go too crazy. I firmly believe in having 5 great credit cards to start and then "garden" and let them age for 2 years while showing very responsible usage.
