We've had several ponies in the 13- 13.3 hand range and all are able to carrie my dad easily (in fact, three have been his). He's probably about 170 lbs, the ponies are of what I consider average build, not refined like the modern shetlands, and not drafty (though we've had a haflinger too, they are characters!) As for a size comparison, if my 31" mini really tried, she could probably squeeze under their bellies. I love this size, pretty much anyone can ride them but they still have all the fun qualities of a pony.
As far as gaited ponies, they can be a lot of fun. We've had a rocky mountain pony and we currently have a gelding who we think has some tennessee walker in him, we also tried a mare who had various gaited breeds behind her. Both the mix mare and the gelding have a normal, two beat trot in addition to their respective gaits, and they can be a bit confusing if you don't know what to ask for, as they kind of make up their own, in between gait if they get mixed signals.