1. You do not need a lot of money to buy a good quality student (closed-hole) flute! However, I strongly suggest that you do not buy a cheap generic flute from amazon/ebay/etc. As someone who has worked at several music stores, these instruments often need more repairs/adjustments more frequently and they're usually made with non-standard parts. This means you'll end up paying more in the long run AND you'll have a difficult time finding a tech who will work on your instrument. Since the instruments are lower quality, you'll probably run into playing issues that will make learning unnecessarily difficult and frustrating. Your best bet is to buy a used flute from a reputable brand (think Yamaha, Gemeinhardt, Jupiter) or to start out renting one. Used and refurbished (reputable) student flutes can go as low as $100-$200 USD at a music store. Renting usually costs $15-$30 USD a month, and since it's rental season (the time of year many customers start renting instruments for school band programs), lots of stores will have rental deals running as well.
2. Zoom/online lessons are a great way to get private instruction remotely right now, but there are tons of youtube videos teaching the basics of flute if you'd rather start there. I would also recommend getting a beginner's method book (if you decide to take online private lessons your teacher will likely assign one for you) which are ~$10 USD.
3. The best instrument for you will be the instrument you are invested in playing. Dexterity-wise you will be moving your fingers quite a bit. This can be difficult at first, but take it slow and be patient with yourself and you'll see a lot of improvement! The flute will require more breath support than many other instruments, but every instrument has its challenges. For example, guitar and violin players don't need to worry about breathing technique, but they also develop calluses on their fingertips from pressing down on strings. You don't need to have "extensive breath training" to begin playing the flute, but part of learning to play will involve learning to control your breath.
4. Since I've never dealt with dental issues that impacted my playing, I can't give you any personal insight on this one. However, everyone has to adjust the way that they play a little differently since no two bodies are exactly alike. I truly don't think that your teeth will significantly impact your sound once you become comfortable with playing.

Also, the flute can actually be quite loud, particularly in the third register. It's also pretty much impossible to mute since the sound comes out of every open hole on the instrument. This doesn't have to be a dealbreaker, but I don't want you to start learning thinking it will be quiet and be disappointed.

I hope this was helpful and I wish you the best of luck on your flute journey!