Entering Tokyo - 2 specific questions
they did away with the rule that first claw damage point enters, correct?


It certainly appears that way. From the manual I'm looking at, there's a note on page 5:


  NOTE: No Monster starts the game in Tokyo. The first player must enter Tokyo City during this step.





  Does this also mean that if you damage the monster in Tokyo during the "Resolve Dice" phase, and they leave Tokyo that you MUST enter Tokyo?


Yes. On the previous page, under "Smash", "If you are outside of Tokyo":


  These Monsters can then decide to Yield and leave Tokyo or stay[...]


Plus the rule from "3. Enter Tokyo" on the next page:


  If no one is in Tokyo, you must enter and place your Monster in Tokyo City.


If you roll a Smash, deal damage to a monster in Tokyo, and that monster decides to leave, when you get around to the Enter Tokyo phase of your turn, there won't be a monster in Tokyo; as a result, you must enter Tokyo yourself.