To save $200 I'd book on Air China, but if it was 100 or less I'd get United like you said. Just be aware that transit can be tough. But if you can wait a little, there are often really good fares to here from Chicago, around 500 is common on Asiana via Seoul. If you do go through United, the 55 min in NRT is fine, their transit process is a well oiled machine.
You could also look at flying to BKK or SIN for a day or two to sight see and scout out if you like the idea of living in BKK. They also have cheap fares to those places often.
I would book a hotel for one night or two in advance, so that when you arrive you are comfortable and can head there straight away without the hassle of dragging bags around while looking for a place. District 1 is a little big, I'd narrow it to within D1 to two areas - I'd look for a hotel either in the "Ben Thanh" area, which is mid-range (see the one linked above), or in the "Bui Vien/Pham Ngu Lao" area, which is the backpacker area but also has mid-range hotels a tiny bit cheaper. After you stay a day or two at that place, you can move to another if in your visit you see another area or hotel you want to try.
Usually the visa address thing isn't a big deal, but you could list a hotel you see online if you wish, or as mentioned book one and use that. Or just use your friend's address.
Re: taxis, uber is available and usually cheaper than taxis. Grab is more common than uber here (competitor to uber), but a tiny bit higher pricing. If you get a taxi, use VinaSun or MaiLinh and you'll be fine.
Re: theft, there are motorbike thieves, who will drive by and grab your stuff. It's an issue, but not a huge deal. No need to worry, just be careful and hold onto your stuff tight. Walking a bit further away from the curb is good advice.
Re: passports in hotels - meh. I give it to them. It's for the police reports, and I don't think I've ever heard of one getting lost. I figure it's safer with them than in my safe anyway. I've even left it with motorbike guys on the street when I rent, although that feels less secure for sure.
Re: time - if you're with your friend, two weeks will go fast and you can visit many fun local places rather than just doing the touristy stuff, of which there is less and is why many move on quickly. But since you want to find a place to stay long term, I agree you could easily get a cheap flight to other places. Many like Hanoi, although I don't like it at all. Some of the smaller towns in the central areas are popular for expats, like Da Nang and Nha Trang and Phan Thiet and Phan Rang/Cam Ranh. Any of those could be a good place to try.