yes, mittens are warmer than gloves. They trap more air and have less surface area to radiate heat.
Not sure why one needs specific "running" mittens, but my best mittens for running in cold come from an outdoor store. My hands are fairly small so I can sometimes use kids' snow mittens. In most cases you'd probably want to stay away from the ones that winter mountaineering expeditions, snow machiners, etc use - and even some ski mittens are too bulky.
We haven't gotten cold this year yet, but I've been using some lightweight ones from REI in warmer temps and Sporthill SP3 ones down to about 20F. They're light enough that I can still hit the lap button on my Polar hrm.
I do have some heavier, somewhat insulated shells that I can slip over lighter ones for coldest temperatures, but rarely use them.
Some people just use socks, but I find them too breezy.