Quadrupeds can carry more weight than Bipeds as stated on the D20 SRD. For a centaur, whose size is Large, it means it can carry three times the weight a human with the same Strength would.
Besides, centaurs inflicts double damage while charging with a spear, as a normal rider would do.
For weapons and armors restrictions, they have a human torso with human arms, so I guess they can wield any M-sized weapon and wear any M-sized torso armor. Your GM might want to prevent you from gaining full bonus of a full plate since it covers only a third of your body. This a pathfinder rule but I think it's a nice addition, although it ultimately depends on the game setting's opinion on human part of the centaur.
Don't forget that centaur is a Large creature, so they have -1 penalty to AC, -1 penalty on attack rolls, -4 penalty on Hide checks, +4 bonus on grapple checks
EDIT : As for the stability bonuses, being a Large creature grants you a +4 for resisting a Bull Rush attempt, and having more than 2 two legs grants you the same bonus for resisting Trip attemps.
EDIT 2 : Tumble has many uses. I'd give no penalty for trying to walk on a slipery floor, but I wouldn't allow a centaur monk to backflip over his ennemies' head. Avoiding opportunity attacks would require players description of attempted maneuvers and allows Tumble check if plausible.