Obtaining missing parts for power armor
There are several ways to get armour pieces and/or frames.
From Exploration
Many of the power armour frames you will find will have randomly spawned pieces on them, and many will be incomplete. According to Fallout wiki, random spawns will be based on level: T-45 power armor pieces at level 1-17, T-51 power armor pieces at 18-30, T-60 power armor pieces at 24, and X-01 power armor pieces at level 26 or more.
There are a number of maps online (IGN, Fallout4map.com) which show the location of all the power armour frames in the game (both random and pre-set) among other things.
From Enemies
You will also find enemies wearing power armour (many of whom are pre-set). These enemies will not drop an armour frame but you can take the pieces off their corpse.
As mentioned in another answer you can take the frame if you make them exit the armour by getting behind them and targetting the fusion core in combat (or reportedly pickpocketing the fusion core before combat). Though apparently if you make an NPC get out of it's frame it will count as "stealing" every time you use it.
From Vendors
If you join the Brotherhood of Steel you will be able to buy T-60 pieces and a frame from Proctor Teagan on The Prydwen (not sure if or how often he re-stocks, but he does restock).
Rowdy at Atom Cats garage will also sell individual pieces of T-45, T-51 and T-60 power armour and a frame (they also have some frames of T-60 standing around you could steal).
You can also buy an empty power armour frame from Commonwealth Weaponry in Diamond City and KL-E-0 in Goodneighbor (if you buy a frame it spawns on the nearby Power Armour service frame).
So Many Frames...
What to do with the spare frames is up to you, they appear to be indestructible, some people have suggested leaving them in lakes or something if you don't want to see them.
Repairing
Yes, broken pieces will be placed in your inventory, and can be repaired and placed back on a frame.
Others have already addressed most of the questions posed. Frames often spawn with random armor sections, and are often incomplete, the spawning is based on player level (T-45s up to level 17, T-51s at 18-30, T-60s at 24 and up, and X-01 at 26 and up.) and frames may not even have a fusion core to power them readily available nearby.
Another answerer has noted that enemies sometimes spawn in power armor, in particular Raider bosses, and when you kill them you can take the armor pieces but not the frame. This is a half-truth. If you kill them while they're wearing the frame, it is true you can't recover it, however if you can get them to eject the frame can be stolen.
Shooting out the fusion core of their armor, which is targetable in VATS and does very little HP damage to the NPC (but takes 'limb' damage easily) can destroy it. The core will enter melt down, and if the person has the time and the room, they'll jettison the core and try to move away from it, before it goes off like a nuke.
If they get out of range quickly enough, or simply survive the explosion, they'll climb out of their depowered suit to continue the fight, and leave the frame available for theft.
As for what to do with spare frames, I've been collecting them. At time of writing, there's 14 of them parked, lined up in neat rows, on the front lawn of my house in Sanctuary. If you leave fusion cores in the frames, NPCs will take them if a fight breaks out. Verified when (spoilers) the townspeople of Covenant turned against me, and stole the power armor I left parked out front of the gate.
Take the cores out to prevent NPCs from taking them, or leave them in advantageous places so settlers in towns you control can use them against raiders. I've heard that after a fight in a town, if you talk to the settlers you can tell them to get out of the armor but have not verified myself.
You can also arrange for humanoid companions to use power armor. Just order them to enter it, and talk to them again to tell them to get out. Best of all, while NPCs need fusion cores, followers do not.
All in all though, you yourself only ever need one frame, so it isn't necessarily advantageous to have more than two frames (one for you and one for your follower) recovered, and armor pieces are easier to store neatly, by sticking them in containers.