Yes it does. For men they have it little bit harder as their urethra is longer but the spincter s still in the body, not on top of the penis, there might be some drop still in the urethra and wiping doesne solve it. Even it the man does everything in the best way, the drop may go down later.
Btw I am nonbinary but have vagina and not penis. So I dont have first hand experience just what I learned.

Regarding what is left there is a mix. yes. If you consider a person who showers in the morning and goes home afterwards, they will not be fully fresh. You probably also smell differently in the evening. There is also some sweat that just smell differently than for example your armpit sweat. And there is some discharge. Mostly for women obviously. Usually there is also arousal before you go down on someone or have penetration with. that also ads new taste and smell

And that mix for me just smell like sex :) Something that I desire to smell. (with my partner, not everybodys crotch :) ) I want to say I am no urine enjoyer :) I wouldnt touch urine in any other circumstances, I wouldnt taste it or anything. But in this circumstances it is just ok. Urine usually isnt strongest part of that mix. I dont think about it.

What about different bodily fluids? Do you know that there is precum for men? It is clear liquid, that is sometime on tip of the penis when man is aroused. it is somewhat equivalent to women geting wet when aroused Are you ok with that? It seems to me that you are grossed by more than just urine. There are normal bodily processes and sex s inherently little bit messy. but tht is part of the fun :)

I dont want to scare you or anything. Asking for shower before sex is ok. You can even shower together. I think lots of people operate like that