Can this be cooked on the stove and Skip oven?

Karen - agree with EmilyC, this can be made on stovetop only.
Figure a 10" frypan with a 2" height will hold about 2.5 liters (a bit over 10 cups).
If you think the recipe will need more volume than that, either cut it in half, or use a saucepan/casserole dish that's larger.
Last, look at Martha Stewart's one-pot pasta recipe (her site, here, other sites) if you want a guide to a stovetop pasta.

Hi Karen, I’ve never cooked it entirely on the stove, but yes, it should work just fine. Just monitor the heat closely to maintain a gentle simmer so the bottom layer doesn’t burn. Or, if you don’t mind dirtying another pan, you could transfer the contents to a casserole dish after simmering for a few minutes in your skillet. Either way should work well!