Is there anywhere that I can just store some equip I dont want on my partners? Like I cleared out the Raider base, will be things be safe there?

In F2 every container, book shelve, fridge, etc. is save to store items in. I think it will be the same in F1, not sure been too long ago but positive.

There is something of a "house" you can get, in the mid-to-late part of the game

Where is it on the map?

Its east of the Hub, or you can catch a Caravan.

However, given that you'll need to go through another location packed with valuable heavy goodies before you're allowed in, I wouldn't bother.

If you're fast, the Gun Runners in LA usually have a lot of cash to buy things with, but you'll need to dodge deathclaw to get there.

Oh, I didnt see the spoiler, Im already in the brotherhood, but I was never given a room.

They don't actually "give" you a room. While exploring the barracks, you'll find some forgettable dweeb who'll introduce himself as your roommate. You have a room. No one bothers to actually tell you about it.