Vertical Drop before venting

Is it allowed for a drain pipe to have a vertical drop after the trap and before the vent or tee into stack (assuming it is within code allowed distance)?
Or would that technically be a S-trap?

Also, I'm faced with another problem plumbing my tub / shower. Because of the way my basement ceiling joists run, I would like to run my tub drain to a 4in horizontal stack that would be about 10ft away instead of the 3 in horizontal stack 2 ft away. There is a main joist is in between the 3in stack and the tub drain, and I don't feel safe cutting into it. And for the same reason, venting is not possible within code permitted distance if I tie the tubs to the 4in horzontal stack. So I was thinking instead of using 2in pvc to run the whole length from tub to 4in stack, would it still be within code to use 4in PVc to run to the 4in stack immediately after the tub / shower traps?

Thanks

The weir of the trap cannot be above the takeoff of the vent so the answer would be no. At least according to the upc

Your question is so convoluted that I am getting a headache trying to figure out how you intend to run the line. The tub has to have a vent BEFORE it drops down, and also before it connects downstream of a toilet.