I run out of cards as others were asking for it using the combo multiple times on me - does this mean I lost? or is drawing multiple action or combo cards allowed in drawing?
It's impossible to run out of cards. The last card in the deck is guaranteed to be an Exploding Kitten. You must either handle it using a Defuse (which would create a deck consisting of just an Exploding Kitten) or lose (which would cause the only other remaining player to win).
If you find yourself playing another turn immediately, you must contend with the Exploding Kitten a second time (unless you Attack or Skip).