im trying to find a set of brother hood T51-b power armor. i already took it from all the paladins in " still in the dark " but i used them all to repair other power armor by mistake. is their anywhere else i can find a dead paladin not in the black mountain crater, repconn headquarters, or near nelis airforce base? or is there anyway i can kill one without losing reputation with the brotherhood?

You could complete the Brotherhood of Steel questline, after which Paladins should start coming out for patrol sometimes when you exit the bunker and you can kill them after that. So far, I've been able to kill NCR and Legion without gaining infamy/becoming hostile by using a stealth boy and then planting a grenade in their pockets, so maybe that could work on any Paladin.

The BoS quartermaster also sells T-51b armour, but not the BoS one i think.

The BoS guy who runs the safehouse has the power armor you're looking for. I ran into him after I destroyed the BoS bunker. He was hostile, obviously, so I had to kill him. I found that his armor disguises me as a member of the BoS.

how soon after i finish the quests do the patrols come out

Get Veronica as a companion and by the end of her sidequest you'll have about 4 sets of it. Not just the "Tb-51 Power Armor" but the "Brotherhood" kind that actually says you're disguised as one of them.

The only other way to get Brotherhood Power Armor outside of the 3 paladins in the locations of Still in the Dark is to visit the home base of the BOS.

There is also BoS patrol around midnight at hidden valley which you can take from.

how do i start her side quest

Talk to her. She'll give it to you eventually.

If I remember right, Torres will sell you sets once you become a Paladin, too.