Your outermost layer of skin cells is relatively safe from this because it's made to be less reactive and protect from outside threats like this (My detailed anatomy knowledge is fuzzy, but my understanding is that you're basically protected by a layer of keratin, which soap does not act on), but yes it does break down animal cells, and this is one of the main reasons that soap is toxic to ingest.

Yes.

Yes. That's why getting soap in an open wound is painful, and why drinking soap is so dangerous.

However, your outer skin is mostly immune to this as it's not made of living cells, but rather a layer of dead cells that made some structures that resist this before they died.