For non payment of rent, Kentucky law requires a 7 day notice, filing forcible detainer, hearing, then forcible eviction. How does acceptance of rent payment in part or whole at any point affect the process?

If they pay they stay.
You are evicting them for non payment.
If they make a payment then that nullifies that they are being evicted for non payment.
Especially paid in full. Partical payment would probably be adjusted retro and could possibly go either way depending on the lateness of the remaining balance.

***I do believe you HAVE to accept payment because then it falls on you that you didn't.

If the 7 day notices was a pay or quit notice, then yes, they have 7 days to pay in full. You may accept partial payments to go towards what they owe, but if they don't pay in full then you fiel for the unlawful detainer. They still have time to pay in full up to the hearing. It will then be up to the judge to set any fiurther deadlines where they can pay and stay.