We were in the Alice and Uluru at the end of August last year. The weather was really pleasant, somewhere in the 70's. We flew to AYQ from the Alice, but our flight was 13 hours late (!!), so I can't particularly recommend this option. The drive is fairly lengthy - four or five hours, so it's your choice. You have a relatively short amount of time to accomplish the roundtrip between the two. There is a large tourist office in the Alice, so they might be able to help you out, but I think you would probably need to arrange things in advance.
In the Alice, the top hotel is the Alice Springs Resort (owned by Voyages, like all the resorts in the Ayer's Rock area). I didn't see anything in the Alice that could match it. It is not luxurious, but is comfortable and well-located. We spent three nights in the Alice, and did not run out of things to do. It is rough around the edges, but has an amazing number of unique tourist attractions - the Royal Flying Doctors, Desert Wildlife Park, School of the Air, and the original telegraph station. We will return.
Ayer's Rock is run like a big theme park. I loved the rock, but hated the atmosphere around the hotels. We were much happier driving around on our own than doing the tours. You can take in the visitor's center close to the Rock for some further information. We stayed at Sails in the Desert, which is generally considered to be the pick of the hotels, but again, you would not really go here for the hotels - except for Longitude 131, which I think has a two night minimum. Going out to see the stars at night, either on a tour or alone, is a highlight, as they are unbelievable. It was nice to have an asronomer pick out the constellations for us. Some of the tours must be reserved well in advance, as they can be booked up - particularly the Sounds of Silence dinner. The Rock is much more interesting than it looks in pictures, as there are canyons and textures that you just cannot see in pictures, but a day will probably be adequate.
With the amount of time you have, you cannot really do justice to both places. I usually prefer to see one place well, rather than two places badly, but that is personal preference. Barring an airline disaster such as ours, the airports at ASP and AYQ are small and very easy to navigate, so it probably is not a bad idea to maximize the flying so you can have more sightseeing time in each place. It is a really amazing and unique part of Australia.