That is quite a vacation/trip!
The Aladdin is a nice hotel. You'll find plenty to do in between the casinos such as shopping (lots and lots of shopping!!!), seeing the shows, eating at the buffets, etc. You can purchase tickets for shows through the hotel which hosts each show. (Example: Blue Man Group - Luxor Hotel).
In LA, there are quite a number of hotels to choose from. My suggestion would be to stay in a hotel near Hollywood. Hollywood is situated pretty much near the center of the "touristy" areas. As far as choosing hotels, it depends on what your budget is. I can mention quite a few of them. The Standard Hotel. The Roosevelt. The Regent Beverly Wilshire. Lots more.
Regarding show tapings, you can get free tickets by visiting the websites for the major tv studios.
http://www.nbc.com/nbc/footer/Tickets.shtml (Jay Leno show is extremely popular)
http://abc.go.com/site/tvtickets.html http://www.cbs.com/info/user_service..._tickets.shtml (You can go on the Price is Right)
http://www.fox.com/community/askfox/ (Scroll down to question #10)
Be sure to submit for tickets in advance. Tickets go quick.