According to the linked ExtJS licensing FAQ, you have obligations to publish the ExtJS JavaScript code that YOU have modified AND(!) you are distributing it. So, in-house projects are excluded by default, as well as all extensions you have downloaded from the Internet - since those are already published back to the community. As I've noted, I never needed to extend ExtJS so far, always found that someone else did what I needed already.
Either case, it does not affect your FPC/Lazarus project parts.
True, that FPC/Lazarus fpweb/FCL-web documentation is not abundant

- not surprising, considering that the latest ExtJS functionality is only in the 2.5.1 branch of FPC and only in the trunk of Lazarus, both from SVN sources, obviously - , but there are some - trivial - examples and demos available to get you over the first few bumps.
For the core Lazarus components you can find demos under
...your Lazarus dir.../components/fpweb/demo
(sessions, how to use templates, etc.)
and for the core FCL parts you can find some demos under
...your FPC dir.../packages/fcl-web/examples
(extjs, json, etc., I'd look at the webdata mini-demos there)
And there are the FPC and Lazarus mailing lists where you can ask your questions from the community or the developers themselves.