Hello Yvette,
Here is my opinion on your questions raised..
Is it bad to put the trub into the fermenter along with the wort? Nope, not if you intend on keeping you beer in the FV for 3-4 weeks. I have not experienced any off flavors, as everything will eventually settle out.
For example, emptying the entire kettle of wort? I have done this before will no ill effect. I usually siphon off the top and leave the last bit of hop residue behind, as this will lead to a clearer beer.
Will it give the beer off flavors? None experienced
If it's a bad move, do you recommend using a secondary after the fermentation stops? I very seldom use a secondary...only for long term aging (Triple), or adding fruit or adjuncts for flavoring.
Is straining the wort the common practice? Yes. I strain the wort when transferring if no siphon is used.
Should I try a cold crash before bottling? IMO this is a good practice. The trub layer will hold tight and lead to a clearer beer overall.
Cheers!