I finished the Fourth Trial first so none of the Temple dudes would attack me once I was named Nerevarine. Since my inventory is getting a bit bloated now thanks to those guys who said they weren't the one in Azura's cave thing, I need to know if it would be safe to put those things away in the Council Club I was allowed to call home after doing the Bad People quest. I my character is more of an unarmored user(I am using the Morrowind Code Patch, which fixes the bug involving that skill) and anything that isn't the Dark Brotherhood armor seems to lower his armor rating, which means most armor is fairly useless to me now. His armor rating thanks to unarmored is 35.

Is there a cap on how much I can store in a house outside the containers? If so, how much? What would be the cap for storage in the Council Club in Balmora? Is it safe to store the items I am getting from the main quest there until I need them?

My character has a non-fortified strength of 100 and is about level 38. He can store 500 pounds of items currently.

It's safe to store items there. It's safe to store them mostly everywhere. More often than not the containers don't respawn.
I don't think there is a cap on how much you can store in a house, but given that you have more than 1000 items stored in the house (items which aren't already stored in a container), an Overlow Loot Bag appears.

You can also store stuff in the crates on the streets of Balmora.

Thanks. I remember reading about there being a cap on how much I could store, and wondered if it was always the same. I'm just going to be safe and make a table out of clothes outside the house. Or make a clothes fort. To put stuff I assume is important in or on, that is.

Well I think it's even safer to leave things on the floor than in containers. I always use Caius's house to store my stuff (a lot of stuff, everything I want to keep but don't want in my inventory, about 400-500 items) and never experienced problems. Some containers can 'eat' your stuff and I don't know the pattern.

If you use the various crates or barrels (which have a weight limit) for all of the lighter items, and just dump the heavier stuff on the floor, it should be a LONG time before you need to worry about the 1000 item cell limit (on console, the PC limit is supposedly higher) and loot bags appearing. My characters often have more than one storage location, with a cave here, a house there, and one of the old Dunmer Fortresses somewhere else, with all of the assorted crates, barrels, and sacks in each of them. Addamasartus, right there near Seyda Neen, is generally the first, and can actually hold a decent collection of stuff for an early-game character (around 6 crates, 2 barrels, a long box, and 2 sacks, if I recall).

Supposedly, the capacity of "perma-corpses" is even higher. If you find either the missing tax collector, Ralen Hlaalu, or the deceased adventurer at the bottom of the stairs in the tomb where the MG orc sends you in the MQ, those should never respawn, and can hold an immense amount of stuff (I've encountered scrolling issues with the list above some point, but the stuff is still there, it just takes more work to access). Other corpses (ones that you create the old-fashioned way) will vanish after 3 days along with their contents, so be careful.