If there was a Gold Bank of Lannisport do you think it would be much of a competitor to the Iron Bank? Notwithstanding the current royal debt, is Westeros even a significant fraction of the Iron Bank’s business? Moreover would many Essosi be interested in traveling to the literal edge of the known world to get a loan from, in their eyes, uncultured barbarians? Would the Lannisters even trust people half a world away to pay them back without their own equivalent to the faceless men?

The Iron Bank does a good deal of business in Westeros, and the Free Cities are not that far away from Westeros.

I don’t think it would be an immediate threat, but more of a long-term issue in terms of their ability to grow in that market. 

What would have happened if Robert had killed Rhaegar at the Trident, but then died of his wounds on his way to Kings Landing? Lets say he doesn’t go until Eddard finds Jaime on the Iron Throne. Would Stannis be made King even though he’s currently under siege and Robert never sat the throne? Or would the rebels need to think of something entirely different?

No, they’d go with Stannis. The same logic that put Robert there puts Stannis there – second cousin to the surviving royals, next in line by blood once Viserys dies. 

A Bank of Lannister would probably have to import Braavosi accountants, no? Suppose those accountants demanded as their price for repatriating to the Rock: eligibility to marry into the Lannister family (because secondary world fantasy about those Medicis is big in Braavos, don’t you know). Do you think Tywin would have accepted on those terms? Would Cersei?

The Westerosi have some form of accounting, so maybe not. I have to say I think the Lannisters would probably deal with it through industrial espionage instead of marriage.