Can a relation be a partial order and an equivalence at the same time? Can a relation be a partial order AND an equivalence at the same time?
For instance, if we have a set $A = \{1, 2, 3, 4, 5\}$ and a relation $R$ on $A$ defined as $R = \{(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)\}$: this relation is reflexive, anti-symmetric, symmetric, and would it be considered transitive as well? If it is considered transitive, I suppose that it is an equivalence and a partial order at the same time.

Yes, but it must be exactly like the one you propoes, that is all the equivalence classes must be of size one.
For if we had an equivalence class with at least two elements a,b we would $(a,b),(b,a)\in R$
so it would not satisfy antisymmetry
