Week in Catalonia Itinerary Check

Of course, everybody has their own travel style, but your itinerary sounds reasonable to me. If anything, I'd add more time in Girona.
In any case, pre-book hotels in advance, and don't leave it until the very last minute.

Rather than changing hotels night 6, you could drive back to Girona for the night. Aside from the Dali house, there’s not a lot more to visit, unless you go see Gala’s chateau, too.
Night 8, you need to stay close to the airport or at least in Barcelona. If the flight is at 9:30, international check in starts at 6:30. Intra-European flight check in would be 7:30. All you need is morning rush hour traffic jam, an accident, car problem, problem at the rental car drop off and you are looking at buying yourself a new ticket.

Thank you both! Are there any places in particular you would recommend for Day 8 night that is closer? Or keep it simple with just heading back to Barcelona for the night.

Assuming you are taking a flight home to North America, I’d stay near the airport at a hotel with shuttle service. By the time you drive there, turn your car in, and get to your hotel, it will probably be late. Second choice would be in Barcelona, but you have to allow time to turn your car in somewhere. The airport will have the widest range of hours that they are open.
Edit: I missed that this is a Sunday morning, so no jams, but still you need to be closer to the airport.

Where are you supposed to drop off your car? Traffic "into" Barcelona on a Sunday morning, especially at those hours, won't be a problem -if anything, people would be driving "out of" Barcelona for the day.

Yet, I'd also choose a departure point closer to the city, just in case... Begur is a long stretch from the airport, and the roads out of Begur are county roads (mostly good roads, but one lane only) until you reach the motorway. If any incident were to happen, you'd be stuck.

Would spending Day 8 am in Begur and then working our way down to Sitges for the Day 8 night make sense? Much closer.

Yes that would make more sense to me. Yet I am assuming you're dropping off your rental at the airport, then, right?

Actually went ahead and booked that last night at the airport hotel. Will let us maximize that day rather than worry about another destination. Thank you to all for the feedback.

Day 8. Drive back to Barcelona and turn in rental car. Stay that night at a hotel near the airport that will shuttle you there for arrival at the terminal at least three hours before departure.

What is "the airport hotel"?... the airport per se doesn't have any hotel -to my knowledge. There are indeed hotels "near"(ish) the airport, but it's not quite "the" airport. Besides, on a side note, those closest to the airport are located in a doggy neighbourhood, just saying. Nothing wrong with booking there, simply watch where you go if you go out for a walk.