Does it matter what city I begin investment banking?

Depends on what you want to do. Plus these days there are many virtual options. But again it all depends. 

It shouldn't matter. If you have a logical story on why you want to work at a specific fund or city you will be fine. If you have no ties to NYC and are trying to work there for PE then it's going to be harder to sell your story if you don't have ties to the area. If you do have ties to the area then you will have to sell them on that.

Depends on what city and group. Houston is tough to recruit out of, there's a real stigma in generalist PE against the oil and gas groups - MF PE is basically unheard of unless you're working in one of their few remaining energy groups, even generalist MM/UMM in NYC is a challenge. SF isn't too difficult since TMT/comms are attractive groups. Chicago is a toss-up and dependent on what group. Plus there are logistical challenges - future on cycle will likely be in person, so you'd have to fly there from your regional city the second you hear about it, likely missing some interviews

Generally if you are set on MF PE I'd recommend NYC. But if you have no better options, can always lateral after a year.