The count happens in a Joint Session of Congress, presided over by the President of the Senate (Vice President Mike Pence). 3 U.S.C. §15 outlines the procedure. When the count begins, Pence will start with Alabama and proceed in alphabetical order.
As each state is counted, members of either House have a chance to object. Any objection must be filed in writing, without argument, and signed by at least one member of the House and one member of the Senate.
If the objection is then received, the procedure says that the Senate has to return to its chamber, and each House decides whether to support the objection (sustain it) or oppose it (overrule it). Once both have decided, the Joint Session resumes. The vote of each House is then recorded.
Can Vice President Pence, in his capacity as the President of the Senate, pick and choose which electoral votes to count? Probably not.
Twice in American history, the President of the Senate has both been a candidate for President, and decisively used his powers during the counting of the vote. In 1797, John Adams counted a decisive slate of votes from Vermont which had been questioned in the spring. And in 1801, Thomas Jefferson similarly counted a decisive slate from Georgia which were legally defective on their face.
In neither case did the opposing party object to decisions undertaken by the presiding candidate. While there is clear historical precedent that an interested President of the Senate may exert influence over the count, his powers are limited, particularly where an opposing party is motivated to contest his decision.