You CAN resize without depriming just take out your decapping stem.
Also, I have NEVER trimmed ANY pistol brass, new or used, so you're prolly fine. I've been loading for a lot of calibers, and for over 10 years with very few problems.
As far as running new brass through your whole process, of course.
I've got a progressive, so it's no extra work for me, but I sometimes resize/deprime, clean, prime, THEN store in baggies, so obviously, I wouldn't want to upstroke deprime those....
I have seen BRAND NEW pistol brass with no flash hole, so I'd definately make sure you got good stuff by running it through your whole process.....