For Hagia Sophia and Blue Mosque, it just depends on day and time that you go to visit. We visited Hagia Sophia later in the day on a week day and the line was very small to purchase tickets and enter. On Saturday, when we walked by, the ticket line and the entry line was rather long. We didn't experience a line at the Blue Mosque, maybe we just got lucky, but there were quite a lot of people.
For the Uber / Taxi to Balat, our CC was accepted and you can pay cash too. The taxi driver on the way to Chora Church did charge more than the Uber price, but not too bad, and the taxi driver on the way back charged exactly what Uber price stated. So just be careful with the taxi drivers.
Caution for some hotel services though. The concierge at our hotel offered a lot of assistance, but they charged way too much money. For example, when we made an inquiry about a ride to Chora Church, he was charging 2,800 TYL - about $96 USD. With Uber Taxi our round trip was about $40 (with one taxi driver over charging). The hotel concierege did the same with other things, charging way too much, so we never used their services.
Restaurants in this area are terribly overpriced and when you walk around, the restaurant staff try to get you to eat at their restaurant. Look at menu prices before you enter and sit down. It is worth the walk to go a couple of blocks out the the tourist area to find really good food at a much more reasonable cost.
We stayed very near the Blue Mosque to save on walking distances because my husband has mobility issues. We paid more for hotel, but being close to Blue Mosque, Hagia Sophia, Basicila Cistern and other venues it was worth it for us.
Overall our experience here in Istanbul was great - just be cautious of people approaching you to try to get you into their restaurants or establishments. When you have a chance to talk with the locals though, we found the people to be very kind.
Safe Travels
Julie Kay