I would like to join the Brotherhood of Steel.
If its not posible to become an official member, how do I get Power armor training?

And also, I've been having trouble with the Holotape quest. I cant get past the Boomers.

If you go their bunker you'll receive a lengthy multi-part quest and afterward I believe you're accepted as an honorary member. There's a second quest you can embark on and the reward for this one is power-armor training.

Ok to get power armor training im pretty sure there are two ways atleast. One is completing all the missions for the BoS and another is to do the quest "For Auld Lang Syne" And to get past the boomers I always sprint to the house with all the armor and stuff on it, then make a straight run for the fence when you at the fence they wont shoot anymore. And they holotape is pretty easy to find once they stop firing at you.

The simple way past the boomers is just to hug the rock wall to the left of the road. You should receive zero damage if you follow the rocks all the way to the fence.

Ok, thanks for the tip. I'll go try it out. (hope I dont get killed again)

This is what I do. I tried the route that guy suggested and could never make heads or tails of it. With any companions left behind and light equipment I rarely take any damage.

Going through the tunnel is much easier, shame it requires such a high lockpicking skill

You can become on official member as the elder grants you the rank of PALADIN in the botherhood of steel.

I gave up on trying to get to the Boomers ages ago. The artillery timing notes are just confusing and I died many times.

Desperate to do the BoS quest, I waited until they stopped firing, then stripped to my underwear (combat armour is heavy), ran to the dead paladin, grabbed the holotape, took some turbo and about made it back in one piece. Not fun.

on the boomers:

1) get to the edge of where they start bombing. hide behind the big rocks to your left/north. the second the bombing stops, run towards the semi destroyed house that is basically the NW most house. the bombs 1 out 3 times might kill you, and the other 2 they might damage you, but if you get into the ne corner of that house, the bombs do no damage...

2) the second the bombing starts again, get out of that house (you have to exit through south part of house). immediately go around the house and head NE towards the fence...they won't bomb close to the fence. I died like 3 or 4 times before instructions similar to these helped me. you can do it!

on BOS armor. if you wear it, i am pretty sure NCR will shoot at you. don't think that is true of remnant armor. with my perks, implant, and just the remnant armor I have a DR of 36. sweet.

if you are doing the BOS quests make sure to do them all at once.
At my save were i wanted the ncr to forger an alliance with the BOS it glictched and the elder would not speak to me only just say ' making preparations to lift the lockdown' this basically ruined my save and what i wanted to do so i would definatly reccomend doing all of the brotherhood quests at the same time to avoid this horrible bug

Ok, I finally did all the quest, and know to oficially join the brotherhood I need to to the Black mountain quest. But I cant find the key to the 2nd floor.
Where can I find it?

Look under the stairway leading up to the second floor.