Road 10 to 11 speed upgrade, what needs to change?

The main question that I have is how do I know which bottom bracket do I need to get for the 11 speed setup to work correctly?

It depends on the crankset. If you use for example a Shimano Hollowtech II crankset, you need a Shimano Hollowtech II bottom bracket. There are quite many cranksets that work with 11speed setup. Thus, it's impossible to give a definite answer unless you have selected the crankset.

You mentioned considering chainrings. If I understand this correctly, you're not planning to change the crankset but rather only the chainrings. If this is the case, the existing crankset will work with the existing bottom bracket -- no need to change the BB.

Typically you would not need to consider the derailleurs (historically they had a standard cable pull ratio), but with the new high-speed setups each generation may change the cable pull ratio. Thus, unfortunately you may need to change the derailleurs as well. At least the rear one.

The rims have no compatibility with the drivetrain speed count. However, the rear hub does.

Anyway, I would suggest to reconsider the "upgrade". It's not a true upgrade in the sense that it would buy you much. It is debatable whether more or less speeds is better (I consider less speeds better because less speeds gives larger jumps between gears). If you for example have 11-30T 10speed cassette, it has 11.79% jumps between gears. A 11-30T 11speed cassette would have 10.55% jumps between gears. Hardly a large improvement. And I would consider this a step backwards rather than a step forwards as I prefer large jumps between gears.

Newer chains seem to be more durable than older chains due to improved materials and manufacturing processes, but it is a good question whether the increased durability of newer chains offsets increased price of the newer chains. Also, the "upgrade" from 10 speeds to 11 speeds costs so much that you're unlikely to ever recoup the investment in saved chainwear costs.

I've done this conversion last year on my #2 bike from a 2013 Ultegra 10-speed to a 105 11-speed, to be able to use the 11-speed equipped wheels of a newer bike on my two bikes alternatively and avoid cassette swaps to do so.

The swapped parts are: the brifters, both derailleurs and the chain. The cassette of course! There is no need to replace the chainrings, the inner width of 10 and 11-speed chains being the same. It is only the outer width that changes because 11-speed has thinner plates. You could also keep the FD but I've chosen the new 105 FD. It is sleeker and easier to set-up and fine tune.

The quality of 105 from the most recent edition is far above Ultegra from '13, due to trickle down of technology. The most expensive part being the brifters.