You should really check out tripadvisor - especially the forums section for LA public transportation advice. Santa Monica is great - I stay there most of the time with DS (3 years). Third Street Promenade is a pedestrian only area with tons of great shops, restaurants and street performers. There's a very, very small aquarium at the base of the pier, there's an amusement park on the pier with a solar powered ferris wheel IIRC, and just a lot going on. You can easily spend a day in the area. Definitely rent bikes or roller blades and head down to Venice on the boardwalk - it's not that far.
I know you said you're not getting a car but I'd spend some time on tripadvisor to get some realistic thoughts about public transportation in LA. It is very slim pickings. If your DH is really dead set against getting a rental car (and I'm not sure why - if you have a GPS, road map and plan a bit beforehand the freeways are not bad at all) then be prepared to spend big $ on transportation and make sure you've got cab company phone numbers and a mobile phone that works without breaking the bank.
The distances don't seem too bad but are very time consuming. I just did a quick google directions search from Universal Studios to Disney and they estimate 45 minutes. I'd allow almost twice that. Frankly, I can't imagine taking a cab from Universal to Disney. I hate taking the shuttle bus from LAX just to my rental car place.
Having said all that, Cheesecake Factory is a chain. Not horrible but nothing to write home about. Red Lobster ditto. I'd use tripadvisor again - they'll be able to help with kid friendly places that are more local. One I've been waiting to try is Hostaria del Piccolo in Santa Monica. Definitely go to In-n-Out - it's a SoCal (now West Coast) burger chain and all they do are burgers, fries and shakes. Google their "secret" menu.
The Grove is good for shopping, Griffith Park, the Natural History Museum, LACMA (you might convince the DCs about going there because they've got the Tim Burton exhibit through 31 October - not sure when you're break is), Getty Center, La Brea Tar Pits, there's a ton of stuff to do depending on your interests and how much time you have. You might also check out latimes.com - the LA Times website.
Good luck and enjoy!