"that which" used together

The combination of that which in the example sentences is fine. The that is a pronoun referring back to a noun phrase and the which is the relative pronoun used for non-animate antecedents. If we expand the shortest of the OP's example sentences to replace the pronoun that with its noun referent, we get:

• A very small part of acting is acting which takes place on the stage!

We can see a similar (personal) pronoun / relative pronoun combination in:

• He who hesitates is lost.

"That which" = What

For example;

We feel great admiration and respect for those who gave their lives for this country. Our people will long remember that which they did/ what they did.

I agree. In this instance, "what" may be more suitable and "approachable". However, "what" does not, in all instances AND grammatically, take the place of "that which". For example: "That which disturbs me, does not necessarily disturb others." VS: "What disturbs me, does not necessarily disturb others". There IS a difference, albeit subtle. IMHO, the first example is much more eloquent than the second, while still comprehensible.