I assume your first question refers to a
Genie+ LL and not an ILL. Anyone, not only those staying at deluxe WDW resorts, can book their first LL at 7:00. You can book your next LL at the earliest of:
1. You tap into your most-recent G+ LL reservation
2. You cancel your most-recent G+ LL reservation
3. Your most-recent G+ LL reservation expires
4. Two park hours (i.e. hours the park is open) pass since making your most-recent G+ LL reservation
If the park opens at 9:00 and the 2-hour rule applies, that is 11:00, not 9:00, when you can book your next LL. Genie+ LL times start with park open. There is no lightning lane during early entry, as I understand it (although I've not been in the park that early to see if they actually let people use it before 9:00). So, your tap in at 8:45 example may not be possible. The earliest return time is 9:00. I don't know when you could book your next one if you did tap in at 8:45 and the 2-hour rule applied to when you could book your next one; 10:45 or 11:00. It would definitely not be 9:00.
For your second question, if you tap into your most-recently-booked G+ LL at 9:15, your 2-hour time becomes reset for 11:15. The 2-hour time is based on your
most-recent booking.
For the first six weeks or so that Genie+ existed, there was a loophole that allowed people to book at the 2-hour time
and at the later tap-in time. That loophole was closed before Thanksgiving 2021.