Usually yes, but check out their Labyrinth implementation which might mean the answer for most features is no.
I just read the white paper, and I haven't seen anything about search, which would require some kind of homomorphic encryption, but I think that's pretty much an unsolved problem: for example, to be able to search message content in Proton Mail, you need to enable the creation of a local search index — it's not done server-side, which means all the messages have to be downloaded locally to be indexed.
Search works fine on WhatsApp, not sure whether it's client / server though.
Because everything is stored locally.
By default yes but you can always disable it
E2EE won't be disable-able
You meant EE2E, a more favorable privacy form than E2EE.