No credit card required
Browse credit cards from a variety of issuers to see if there's a better card for you.
More than likely, yes.
If you review the "terms' of your down payment assistance you might find out that it's a "silent second" mortgage lien that is subject to a Recapture Tax which if you sell the property or refinance the loan there is a "pre-payment penalty" that could be due until a certain time period where it's likely to be recaptured or repaid to the provider. If you received a "grant" then that money wouldn't need to be repaid but often times these DPAs come with some of these strings attached.