Shortribs stovetop??

You can do any sort of braise on the stovetop. Depending on your pan and your stove a flame tamer may be necessary- if you don't have one a cast iron griddle or skillet (if you have one large enough) makes a good substitute. I haven't done anything that would pass for a scientific test- even by the extremely lax standards of "food science" - but my subjective impression is that oven braising is more effective at breaking down the fibers in a piece of meat; the collagens between fibers are relatively easy to break down but if the fibers themselves aren't weakened braises can tend to be somewhat more stringy.

I think so, and agree with Nancy that you'll want to keep an eye on the liquid level. And you'll want to make sure you're not simmering too high -- to convert all that collagen to gelatin you need a low heat (160-180F) so you may want a flame tamer under your pot.

Two ideas for you...
Yes do all on stovetop, but you may need to add more liquid (broth, beer, wine or water) to make sure the pot doesn't cook dry and burn the meat.
Or, if you have a barbecue grill with a cover (that you can get to through the snow), you could put the covered dutch oven on the covered grill for about the same length of time called for in oven recipe.