They have a lot of reliability issues but they are generally minor annoyances, caused by sitting a lot, or come from having the crap beat out of them...nothing you're not going to get with any other machine as capable...
Engine size depends on the options during purchase. The base gas engine is a 620cc v-twin liquid cooled gasoline engine. RCW swears by his 3010 4x4 diahatsu diesel although he has had some issues with their crankshafts on the older models I think that's been resolved on the newer models.
They'll run about 30 mph safely. Anything more and you need to get a dune buggy not a mule...
A mule is an excellent addition for towing, hauling, and general utility use. Light trail riding it will also excel at, although as shink mentioned, they are not too much on ground clearance. There have been some attempts at lifting the mules to get more ground clearance and larger tires on them but with any modifications, they shorten service life. Lifts and larger tires wear out bearings, ball joints, and other steering components at an alarming rate
The mule 1000 was a strange beast... sort of a mix between a motorcycle and a golfcart. The mule 1000 used the engine out of a Kawasaki LTD440 cruiser motorcycle complete with twin vacuum carbs (which are apparently a sore point with most mule 1000 owners as they are VERY expensive to repair) mated to a mule-like transmission.
Apparently they had the possibility to be VERY fast if properly modified, although apparently equally as unsafe as the new ones much above 35 mph....
The mule 1000 had a few model specific problems as well as the normal host of mule general problems. I've seen a few posts lately about shift forks in the mule 1000's. Those apparently give some trouble, as well as the aforementioned vacuum diaphragms inside the carburetors.
Then there's the host of mule problems... Sticky or ineffective drum brakes, grinding while going into gear, steering unresponsive/makes noise, general creaks/cracks/moans/groans, and carburetor gumming/blockage issues.
If you are not going to be using the machine for the utility it was designed for, perhaps you should look into the Teryx. It has the 750cc v-twin engine out of the Brute force and the suspension and brakes to go with it. It would be MUCH more capable off-roading and playing in the mud.