What does the summoner skill in mage tree effect?
 

Does it help repair bots repair ?
Does it help phantom ?

It helps any ability of a unit that summons other units, and also any spell you cast that summons a plurality of units. It has no effect on resurrection effects like 'repair' or 'resurrection'.

So it affects demon portal spell and Demonologists summon skill?

And what about Mystic Egg? It is affected?

I think it does. It affects everything that matters with summon units, even pet dragon and units that summons, like demonologists, royal griffins, royal torns, etc.

It definitely effects demon portal and demonologists... but I am not sure about the egg. If it did, you could get something like 145% of your leadership in an egg stack, which would be cool.

I can't recall testing this one though.

Yes, and it affects also Phantom spell. It is great! :)

Phantomed units can use their ability again. I explained this here.
http://forum.1cpublishing.eu/showthread.php?t=13315

Does this talent affect the Phoenix in any way? From my quick observations I don't think it does.

no it does not
since phoenix is single unit spell just like book of evil


it effect call of nature/demon portal/necro call

it helps summons more from demonologist/royal thorn/dryads/royal griffin/necromaner/thorn warrior/thorn hunters/demons

it does increase the numbers being spawned:)

Every level on the summoner skill increases 15% of the total summoned troop.
It affects royal griffins, demonologists, royal thorn, thorns and dryads but it has no effect on necromant or demons.