Haflinger. It's used for all the things you mentioned above and is a very reliable and sturdy breed.
Like a few others said Quarter horses really do just about everything! I know a horse that is a reining-bred QH who is a barn favorite at rated hunter/jumper shows (although I have heard he has a weird hatred for cows...)!
Quarter horse or Thoroughbred or quarter/tb mix
Small horse/pony breed: Fjord
Normal/big horse: TB
Tb/Quarter cross, Appendix Quarter Horse, Thoroughbred (mostly used for english but can be excellent for western usually ) Half-Arab,Paint.
I have a few more i just need to remember the names of the breeds