My gelding tore his suspensory in the paddock, and even though he could not weight bare on that leg and was confined to a stable 24/7, he was canter pirouetting in his stable on 3 legs, and trying to trot around the stable. He would toss his head and neigh when he saw me coming each day. Seemed totally happy in himself. He ended up doing more damage to himself despite being in a cast, and tore a tendon in the same leg, hence being put to sleep.
So yes, they can certainly still run around and be stupid even if they are on 3 legs!
I would be very cautious about working your horse. My boy was a thoroughbred that had 32 starts under his belt. Before his did his suspensory, he also had a hock injury which was a result of his long stint on the track. He was never truely 'lame' on it, just a bit off, and would get quite grumpy under saddle and when being tacked up. People said he was just being a typical ottb, but I got him xrayed and sure enough, gut instincts were right - arthritic hock in the process of fusing, with a bone spur attached.
If someone had him, that didn't pick up on a subtle change of personality in him and in his attitude to work, I'm sure he would have been pushed through, probably beaten up a bit for being resistant etc. He never bucked or did anything 'naughty', he was just a bit grumpy and behind the leg.