Partial Rent and Back Rent
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"What's the best way to track partial rent payments if a renter has to break the monthly payment into 2 payments at times?"
The simplest approach would be to locate/select the "first" rent transaction for the period in its Quicken register, then use Copy Transaction/Paste Transaction to put a copy of the rent transaction in the same register. You can then modify any of the fields in the Copied transaction to suit reality and Save it.
" ... if a renter pays me the June rent in July, is there a mechanism to record it as Junes' rent, other than adjusting the notes field to put in a note?"
For Quicken to apply rent to the correct period, the rent received transaction must have a transaction date that falls within the period the rent is due.
Create an "undeposited rent" account (a Quicken cash account will do).
Record your rent reminder in the undeposited rent account with a transaction date that falls within the period the rent is due.
When you enter the actual rent check in your Quicken checking account, use the actual transaction date and categorize it as a Transfer from the undeposited rent account.