September is a fantastic time to go to Vegas. We've been 6 times in September, then last year we couldn't go at that time so we went in June. September again this year though! It's lovely and warm!
As for locations - I don't know what you will be offered on timeshare - I know there are several, but don't know if they belong to the same 'chains' etc. The most central one I can think of is Polo Towers, which would be a good location. Jockey Club is also timeshare I believe but is right next to the construction between Bellagio and Monte Carlo. I suppose Polo Towers faces that same construction anyway.
Anywhere on or close to the strip between Tropicana Avenue and Sands Avenue/Spring mountain road would be considered central. You will probably have to look at a map to see what I mean by that. Basically MGM - Palazzo/Wynn is central.
As for hire car. Don't know if you would need or want one. We have been hiring a car the last couple of years, but only for a few days, and have also made this when we are staying off strip (at Westin and Orleans). Both times I have booked with Airline Network
http://www.airlinenetwork.co.uk and the car has been from Dollar at Treasure Island. They cost about £20 per night including tank of fuel. You add the full tank of fuel for about £5 which is a bargain, and it also upgrades your insurance a bit - I can't remember the name of this upgrade but it will be clear on their website.
As for flights, yes they can be tiresome. Some people like direct, but to be honest I struggle for 11 hours on a flight anyway and like to stretch my legs. We had a 4 hour stopover in Chicago one time and actually quite enjoyed it. Just had some lunch and some wine and before we knew it we were on our way!