Spanish "Jaime" and "Diego", do both translate roughly as "James"?

Factual Questions
1

The name Jaime certainly seems most like the English name, but on the other hand I can see that “Diego” descends from “Jacobus”* through variants such as “Iago”. But how did Spanish evolve the variant “Jaime”–and how did we get “James” in English, in the first place?

*“Jacob” being the original name from which “James” descended.

2

Latin: Jacobus
Late Latin: Jacmus

And I’m sure you can see how Jacmus became James and Jamie.

3

Some info, but it basically says “no” about “Jaime.”