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The existence of the Frobenius endomorphism probably goes back to Euler's proof of Fermat's little theorem. But why is it named after Frobenius? Who gave it this name? When was it first stated in full generality? How did people refer to this concept before the language of ring homomorphisms?

This is traced in Hasse (1967) and Hawkins (2013), who writes on p. 326:

According to Miyake (1989, p. 347), Hasse introduced the term “Frobenius substitution” in (1926-1930), apparently unaware that Dedekind had discovered it independently and probably earlier.

Since you reach back to Euler, who proved Fermat's little theorem in the form $a^p equiv a bmod p$ by using induction on $a$ and the binomial theorem, I think your "Frobenius endomorphism" is the $p$th-power map in characteristic $p$ (or $p^k$-th power map if the base field has order $p^k$).

Frobenius has his name associated to this rather elementary operation, used long before him, because he proved the existence of lifts of the $p$-th power map to finite Galois groups over $mathbf Q$. Those automorphisms, which Francois mentions in his answer as Frobenius substitutions, are more intricate than the $p$-th power map in characteristic $p$ and it is not surprising that the person who first published a paper about them got them named after him. (Dedekind wrote to Frobenius that he had gotten the existence of such lifts to Galois groups over $mathbf Q$ earlier, but this was just a private communication.) Once the name Frobenius substitution was used, it is not surprising that the map in characteristic $p$ got named after Frobenius too. However, I don't know who first used his name for the characteristic $p$ operation.