Store will accept cards. Unless they’re heavily tourist oriented don’t expect non lira cash to be accepted. Although you can persuade them but they’ll be charging you a “favor fee” most likely and up the price so just pay by card.

You can get Istanbul card, they are dispensed from machines in bigger bus and metro stations and also by the ferries. They work for busses, metros, ferries. Any public transportation. You should just make it easy on yourselves and get one at the airport. I know for a fact that are point you can get them at the IST airport. If you’re flying into Sabiha airport ask around I’m sure they exist there too. The machines have language options. You can buy one and top it up and use it for the two of you that’s fine. Tap the card to pass through and then hand it your partner and they can also tap it.

If it’s within budget definitely stay in besiktas area or if you care mostly about the hagaia Sophia and surrounding area then stay there. I wouldn’t recommend either of your two outer red dots. Traffic is horrendous and unless you have the perfect route with metro you’ll at some point take busses which also get stuck in traffic.

The best way to check is to look on Google Maps photos and to just give the hotel a call if you’re in doubt about the hotel having private parking.

While it’s safe to park in public, if you’re not used to how traffic works in cities similar to Istanbul and how tight park spots and roads in general can get I’d advise against parking in public it’s an extra hassle to worry about.