Cannot find slow (frozen) running query in Query Store
Not a full answer to all your questions but:


  What reasons makes the query stop from increasing CPU time?


As well as heavy IO (though I would expect CPU time should slowly tick up in that case too), there is the possibility that it is sat idle waiting for locks to be released by one or more other transactions. Check the relevant output of sp_who2 (built in but undocumented) or sp_whoisactive.


  I have heard about trace flags and extended events, but I do not have any experience with this.


Trace flags are not something to play with without a specific and well understood need. Extended events will be useful for tracking plan execution amongst many other things, but a full run-down is almost certainly out of the scope of a DBA.SE answer. There are plenty of examples out there, and presumably official documentation (though that tends to be optimised for reference, not initial learning).