Is there a deadline by which votes must be counted prior to this House procedure?
There is no "voting for President" in the US Constitution. Voting is not necessary for the appointment of Electors.
There are statutory deadlines in 3 USC, Chapter 1.
Counting electoral ballots starts on Jan 6. (3 USC 15)
Sttes can confuse the process by seldning more than one slate of ballots. I expect DEM states to do this, send DEM electoral ballots even if Trump wins the state, based on "popular vote" or some other rationale.
There is 2 hour time limit for debate (3 USC 17), but that doesn't resolve a situation when the vote is tied.
I think the reality is that the US is no longer a regime of orderly transfer of power. We literally are a banana republic.