Title says it all. If you don't remove and clean the strainer, will old oil be left in the engine?

Not much oil, but you should clean the strainer. Depending on the state of your engine, you can learn a lot from what's in the strainer and what's on your sump plate.

Don't over torque the sump plate!

You don't have to.. But you should want to.
All the keys to engine health lie in the screen and sump plate, where the debris settles between service intervals..