No the factions don not have access to a general-training building (unless you're playing a modded game...)
What i gather from experience as a big user of assassins (My king is always This or that the malevolent or the tyrant) is that ther e must be a minimum cap to the number of generals the AI-controlled factions must have.
The cap must be around three, ie the King, heir and one other general...(I thing it was already like that in MTW and RTW... STW is too far for me to remember...)...or King and two generals ...
So if at the end of a turn you've killed all generals except the faction leader, the AI will automatically spawn one or two new generals raised from the captains as you can see in their names (generally if it was Captain Mucho it will be general Mucho Gusto, f.e.).
In order to get rid of a faction this way, you have to get rid of all generals, heir, faction leader in the same turn...(I'm a big fan of that... I've done it to the stinking Milanese with one assassin for the king...and a trebuchet shot for the heir-become king...the last general died in my cavalry charge...
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Hope that helps...
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