1) Nothing. Gore could have (as Nixon did) counted the votes for his opponent (Bush), or, he could have, as Jefferson did, counted the contested vote for himself, or, he could have (as Adams did) have allowed time for an objection from the floor.
There is NOTHING anywhere, any place that says that the veep/pres of the senate cannot entertain an objection from the floor during the counting. Indeed, quite the contrary, his very authority as “presiding officer of the senate” means he is REQUIRED to.
I answered your second question. He acts DUALLY in a role of veep/counter and as “presider” and is fully empowered to count what he wants, entertain objections. whatever. It does not say, “and in that role he must count only those electoral votes given to him and may not count, reject, or entertain any other electoral votes.”
And 1876 proves fully that this entire situation is fluid.
The Constitution says nothing about entertaining objections from the floor, so that would have to be a matter of statute. That's the only role the process in the 1887 Federal statute plays here -- and I have said here on several occasions that some key provisions of that statute are clearly unconstitutional.
Interestingly, once an objection from the floor is "heard" and recognized by the VP, the VP has less power than anyone else in the room -- because he can only vote on any matter if there is a tie in the Senate chamber.
... and is fully empowered to count what he wants ...
Can you cite a single shred of evidence to support this -- either in the U.S. Constitution or in any Federal statute? Maybe it's just me, but it seems absurd on its face.