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No. Banks don't affect each other. The point of getting more banks is to get more money when you cash in. Also T4 bank increases your max amount even if you don't use the ability. A 230 bank takes ~10 turns to hit max. IMO, if you don't have the +2500 knowledge trait, you should cash out around ~6.8k / before hitting 7k (max) (~9 turns) at the start of a round to avoid wasting a round for only ~$200 profit. This also gets you a small head start to build the investment for the following round.
However if you are playing long term, then monkeyopolis with 4+ 420 farms is better. Some maps are open enough to get 8 or 9 farms (with the radius from a 204 village pre-T5). Make sure to have a 205 farm somewhere else on the map.
If you use the Ben hero, you gain interest faster, although I don't know if it is worth it because of his downsides (trojan). If you sell Ben to keep him lvl 6 and lower, I think you can get 12% bank interest at level 4 or 5? Otherwise you need more banks to offset his trojan which reduces RBE and thus money.
The best of course is Merchantmen but that requires considerable investment and planning with 30+ MM type buildings, and you'd only do that if you are going for a big temple or similar (ie. $2mill+ investment).
yes this is true, each bank handles separately, but you do have the option to "collect all" also, which will collect all the money from all the banks or you can collect them individually, all banks have the same interest rate for their own bank