Does the fall dectection work when on water 💧 protection mode

That is correct. See the 'Can I go swimming or take a shower with my Apple Watch?' section in:

About Apple Watch water resistance - Apple Support

which says:

"Showering with Apple Watch Series 2 and newer is ok, but we recommend not exposing Apple Watch to soaps, shampoos, conditioners, lotions, and perfumes as they can negatively affect water seals and acoustic membranes. Apple Watch should be cleaned with fresh water and dried with a lint free-cloth if it comes in contact with anything other than fresh water.

Water resistance isn't a permanent condition and may diminish over time. Apple Watch can't be rechecked or resealed for water resistance."

Yes and yes.

But unless you are inside a pool, there's absolutely no need to engage Water Lock. Disengage water lock before getting out of the pool in such cases.

Outside of that, you can keep Water Lock off.

Water lock has no effect on any part of the watch beyond muting the touch screen input. Water and touch screens don’t work well together, so the water lock “mutes” the touch screen. The water lock also ejects water from the speaker cavity when switching off the lock, though that does not otherwise change the water resistance.