will mining centralize more (maybe in the US?)
The mining centralization is a big discussion. I believe that it won't have a standard increase or decrease, so I highly doubt a 51% attack from a pool. Even if a pool owns the majority of the computational power it won't attack the network, because it won't be beneficial. Although, the whole point is ruined, then, so let's praise that to never happen. Can you imagine the FUD? I guess that once the miners see such case, they'll switch the pool the work on.
When we talk about "bitcoin transactions", will we be referring to on-chain, or lightning transactions (i.e will lightning be more relevant to the avg user than on-chain)?
Bitcoin can't work if we broadcast each transaction we make. It's practically incorrect. The Lightning network must sooner or later be applied for making the whole thing practical. In the end of this decade, assuming that Bitcoin has reached $10M, which doesn't seem impossible to me, a transaction may worth hundreds of dollars to be made. The average user will have to be adapt with it.
Are there any contentious BIPs, that could force another block-size type war?
None that I have in mind.
If there are none currently, what part of the protocol do you suspect may eventually lead to the next hard fork?
The problem with secp256k1 will surely create a hard fork. I just don't know if it'll be the next one.