Some exchanges that do support dollars and withdrawals to your bank account are GDAX, Bitstamp, Kraken
Bitfinex also supports dollars, but you can't withdraw to your bank account. What you can do is withdraw the dollars as tethers. Whether you can use those tethers for anything is another question.
As I stated in an earlier post, Bitfinex is different from other exchanges; on Bitfinex, USD and USDT are equivalent. Whereas other exchanges have a trading pair for USD and USDT, Bitfinex makes no distinction between US Dollars and USD Tethers. Which means that if you have USD on Bitfinex, you can withdraw the same amount of USDT without having to make a trade to convert your USD to USDT.
That is because Bitfinex owns Tether.
does tether have its own wallet that you can store it?
where you'll find a web wallet you can send tethers to. Supposedly you can withdraw the money to your bank, but I've never done it and I have no idea if it works or not, especially considering that tether's parent company, bitfinex, has lost its banking connections in the U.S.
The omniwallet is an offline wallet, but again, I have never used it. I encourage anyone who's interested to explore it. Just now I tried to go to omniwallet.org, but the site is down.
The omni blog is here: https://blog.omni.foundation/