I too had this problem a year and half ago… when I had to choose a framework for all my future needs… as you can see I chosed Yii and as the saying goes… "never looked back"…
It’s difficult that you will get an objective response here (as it would be on any other framework forum)… if you post the same question on all PHP framework forums… you would get 98% always the same response - yes the X framework is just for you…
So if you want an objective answer it’s better to post that question on an general web development forum (not on specific framework forum)…
On the other side, like I had done… you can read all the frameworks features… abit of documentations… then choose from that at least 3 frameworks that you will inspect a bit more… for that 3 try the demos… try to google for that frameworks… and you will get to the framework you need…
and when you choose the framework, make a last test to be sure it’s the right one…
try to do an example project - build a new app with some functionality you will need most in your future projects… so that you get a first hand experience with the new framework… try to ask a question in the forum (to see if the framework community is active and helpful)…
At least that is what I had done at time I needed to choose
And now a subjective answer to your question (sorry can’t be too objective )
[b]
Y[/b]es it is - Yii is right for your need
Yii can do all you need… I haven’t used extJS but here on the forum one user is very actively using extJS and contributing some extensions to facilitate use of extJS with Yii… you can search for that on the forum and get in contact with that user… maybe he can share some experience with Yii/extJS with you…