We see mages like Jaina who is primarily a frost mage, sometimes uses arcane, but rarely uses fire.
Then you have someone like Khadgar who we see frequently using all three schools.
It stands to reason that all mages know at least a little bit about the three schools, especially with the class design revamp and how we see people throughout lore use them.
It would stand to reason that archmages are capable of shaping the arcane with much more skill, and can more easily shift it into frost, or arcane depending on the need for the situation, so I would assume yes.
Anyone who’s actually worthy of being considered an Archmage would be able to use all schools but that doesn’t mean they can’t have preferences.
Jaina melts if she uses fire magic too much, she is an ice mage after all.
There’s not much we know about the Kirin Tor and their process of educating Mages tbh. I would expect that all Mages of repute can cast elemental magic, some just specialize like players do.
Khadgar is in a unique position as he was poised to become the last Guardian, meaning he’s kind of expected to have a grasp on almost all forms of magic.
EDIT: AFAIK Arcane is considered the basis for all Mages and then they transition to elemental magic, as Arcane is more stable and predictable.
The smartest mages don’t bother with fire and frost because they know that Arcane magic is the strongest. 
No, they become warlocks because they know that Fel is the strongest.
Fel is stanky, trust me. Arcane > all other schools of magic.
You’re supposed to put it on a leash and throw it at people, not inject it into your eyes. Canonically, Fel > Arcane.
I tried eating it and it tasted so bad. Not to mention that it smells like sulfur. 
Wait are you telling me lock cookies aren’t real cookies? This is too much.