Talk to Lazlo in the Lower Ward marketplace. You'll learn that maybe you really don't want to get inside after all.
The Unity of the Rings. Go back to the slab where you woke up. Speaking of the Unity of the Rings, I hope you got your mage training from Old Mebbeth and learned some stuff about her from Ravel. Be sure to talk to Old Mebbeth again when you're back from the Planes.
Solving it just makes it more valuable so you get more when you sell it.
NEVER LET GO OF THE SPHERE! *THE* MOST IMPORTANT ITEM IN THE GAME.
Perhaps you'll figure out how to make use of it in the Fortress of Regrets. More specifically, you'll find out in the Maze of Reflections. If you finish the game without figuring it out, then post back here. You must learn its secrets eventually.
You can use it to piss her off. Worship/mock/whatever it enough times and you'll incur the Lady's wrath.
Nothing but a dead end. As I remember Undersigil was supposed to be a huge dungeon with a quest or two. Probably ties in with that solitary sohmien you find down there.
Greater Glabrezu will occassionally spawn down there after your Planar tour. Most of the times you'll get some 300 odd copper as treasure, but sometimes you'll get powerful magic items like Ring Zero and Bell's Shield.
Once they're equipped, open the context menu and select the Use Item button. I think it looks like a hand holding a key.
Play with it.

Abuse it and it will run away. It dies if you take it to the Fortress.
The practical use requires some tricky navigation of a dialog sequence and some running. You can get Marissa from the Brothel to turn your lim-lim to stone. Take the statue to Giltspur in the Lower Ward and sell it to him. He'll give you a password to the Anarchists in the Warehouse so you buy from their stores. You don't have to be in the Faction to buy from them. Joining the Faction is done elsewhere. One thing to note is that their stores are the only place I've been able to find another Cloudkill scroll. Having Dak'kon or Ignus packing a Cloudkill in addition to the Nameless One can be really handy.
The trickiness is in getting Marissa to stone the lim-lim. Due to a scripting error, it won't trigger unless you ask her to go first when you first meet her. If you did it correctly, then it will show up whenever you have the lim-lim in your inventory.