[mivayan]

I have just knocked down the walls of the Machaka Lich Lord's Mausoleum. If I now storm the castle, and he sends his large army into the province from next door, will I:

You will most likely:
First fight the large army + the lich lord.
Then fight the lich lord inside the castle.

Since he is immortal he will respawn after he dies in the first fight as long as he has positive dominion.

Also, once I kill the lich lord in his home province, does his army disappear ala the orc hordes who vaporized after Sauron was killed, or do they remain, but leaderless?

The lich dying will not effect his army at all, and even if it dies in its home province it can be called back by priests.