The universe is finite. It is constrained by laws of nature and it has a limited amount of space (even though it is expanding). When one tries to explain and understand the origins of the universe, one always backs into an infinite regress. For instance, if one explains the existence of earth through the Big Bang, it is not clear where the original matter necessary for the Big Bang came from. Could one prove the existence of an external force that is the origin of the universe based on the above facts?
In the narrow sense, the answer is yes. Consider the decimal expansion of pi, 3.14159.... It is an infinite series of numbers, but you can derive any number of finite sequences from it (1415, 314, 159, and so forth).
In the larger sense, this is analogous to the problem of evil (how can an all good God create a world that contains evil), in as much as it is a question of how a perfect being could ever create anything less than perfect (where the finite would be considered as imperfect and the infinite as perfect).
As such, I'm not sure it has a definitive answer, but you might look up considerations of the problem of evil to see how great minds have tackled it.
It is not known if the universe is finite or infinite.
Although we assume that the laws of physics apply uniformly in our observable universe (viz. homogeneity and isotropy), it is not know if the same rules apply in causally-unconnected parts of our universe. By causally-unconnected, I mean those areas beyond our observable universe, so far away that light (information) has not had enough time to reach us.
Big Bang Theory requires no "original matter". Space, matter, and energy were all created in the big bang event. Our universe if often described by supporters of big bang (almost everyone) as the ultimate free lunch. I believe the technical mechanism preferred by cosmologists is some sort of quantum fluctuation where energy is borrowed from "the future". That sounds wacky, but so does most quantum theory.
So, in reply to your headline questions, I would say yes, I believe that most cosmologists would be happy to say that our universe is "self-originating".
EDIT I should have not said "matter" was created in the big bang event. Only space and energy were created. Some energy would then have become matter when things had cooled down enough. (Recall that Einstein's famous equation declares that energy and matter are the same thing in different forms - E = mc2.)