1) Yes, if you have the original disk(s) or download codes.
2) Your saves should be fine after reinstalling everything (including your Store items).
3) Yes, I believe the original purchase allows for installation on 3 different computers.
4) Your families should still be available as long as you've backed up your entire game on your old computer and restore the files to the new computer.
Be sure to check the specifications of the new laptop carefully to make sure it will easily run Sims 3. They aren't always the best choice to run the game, although you'll likely see several people disagreeing with me. Laptops can be bad about heating up and need to be placed on a laptop fan to keep them cool. I don't know if this is a universal problem with all laptops or with just some of them. Laptops in general aren't meant to run high power games like Sims 3. They often don't have the graphics technology to run the game properly. Other members will be along soon who can explain the details much better than I can.