This is not dictated by any bitcoin ATM for sure, generally you may run this business without having a bank relationship at all. But then you need to convert your cash to bitcoins in some way on regular basis. E.g. if you find a miner that will be ready to sell you bitcoins for cash - then you just found a workaround. There could be other examples. The ideal case with having bank account basically means you get exchange feed with a minor exchange fee (like 0.2%). In case of buying bitcoins in other places - you need to negotiate the price. E.g. miner will want to sell you bitcoins at higher than exchange rate, because he will need to somehow deal with your cash later on.

According to my discussions with operators - to find a bank, that will know about your bitcoin business and still continue relations - is very very hard.
It is not only about "bitcoin" which banks resistant to, but also you will deal with a lot of cash, which makes it even harder to persuade the bank. They just treat you as too risky and close accounts.

In Australia there was a case, when machine operator was fully AML/KYC compliant (obeyed all required laws), but banks didn't want to have such a customer.

Generally, as it turns out in North America (US, Canada) it is pretty similar nowadays.

But again, this is not required by any machine. As normally every bitcoin ATM software supports both: operations from your hot wallet (in this case you might no even connect it to exchange, but replenish manually), or automatically repurchase coins from exchanges and get deposited to your hot wallet. It works like this: you have some liquidity in your hot wallet, whenever customers buy bitcoins from you - there is a mirror deal on the exchange for the same amount.
It is better because you do 2 deals immediately, and get your fees in a guaranteed ways. Otherwise (manual replenishment) - you hold the exchange risk, because after you purchased bitcoin the price might go up or down before customer purchases from you. In case of exchange mirroring you bare the exchange risk only for the amount of your liquidity volume, which is much lower.

Connection to exchange is usually just a setting which you need to do in the back office of the machine one time. Different bitcoin ATM manufacturers support different ones, but the general ones are usually supported.
As an example, go to general bytes site: http://www.generalbytes.com/batmtwo/

And you see supported exchanges:
Bitstamp.net   Yes
Coinbase.com   Yes
Bitfinex.com   Yes
TheRockTrading.com   Yes
Cointrader.net   Yes
Quadrigacx.com   Yes
Kraken.com   Yes
Digitalx.com   Yes

Also there is a list of supported wallets:
Blockchain.info   Yes
Coinkite.com   Yes
bitreserve.org   Yes
Coinbase.com   Yes
Bitcoind   Yes
Litecoind   Yes
Dogecoind   Yes
Block.io   Yes