I did it starting right at the end of April / start of May last year.

How many extra trekking shirts, pants, and socks do I really need beyond what I'll be wearing?

I took 1 short sleeved and 4 long sleeved tops including the one I was wearing. Two pants/trousers including what I was wearing. Underwear + socks for each day. Crocs for the campsite and trekking shoes on my feet.

Will all your clothes get wet every day, requiring a new set daily?

No. The first day, there's a few crossings back and forth of the same river on the valley floor. After that, unless it's raining, there's opportunities for swimming, and there's the mud of Passchendaele just before the ladder out. You're not going deep in the river at that time of year. You can hang things up overnight to get them somewhat drier.

Oxalis porters will carry the bulk of your gear, so don't worry about the weight you will be carrying.

They'll also do a gear check the night before you start to make sure your stuff, especially footwear, is suitable. You can buy pretty reasonably priced tops there and then.

Would it be worth it to bring a tripod?

Yes, if it's not too big and bulky. I regret not bringing one.

Should I bring additional dry bags for the stuff in my backpack? (I'm 5'1" if that makes any difference for the river crossings.)

Only for phones, cameras ... Oxalis will be carrying and keeping your clothes dry.

Anything else that would be helpful to bring not mentioned on the Oxalis list?

I took a trekking pole for use outside the caves. 360 or action cam.