I suspect that one could as easily say "Ili varme salutis."
Not if there are quotation marks.
ili elkore diris "Saluton"
Why would this translate correctly? The phrase "warmly" means the same thing(friendly) in Esperanto and English? That seems strange.
Because Esperanto is awesome! Esperanto is a constantly developing language, and it is constantly taking in many things from many different languages, English included.
Shouldn't warmly be vermete instead of varme?
No. Vermete means "as/like a little worm."
But then why does varme mean warmly instead of hotly, if varma means hot and varmeta means warm? How would you say hotly, then?
Off the top of my head, I might consider "arde."
But this is a hot topic and deserves some ardent discussion.
What I think is (in the case of bath water)
Varmeta = a little bit warm (35°C)
Varma = warm (38°C)
Varmega = hot (42°C)
So hotly should be varmege