Some people put pipe dope on plastic threads, others don't. If the nuts are hand tight (maybe a bit more with a tool if needed) none of your joints should leak. The washers do all the sealing for the most part.
Hey there! You probably already got it up and running. I just returned from a trip to Glacier National Park.
Looks good to me. I would just hook up the water and try running it. I typically don't use dope or tape on a p-trap. It works best just to run hot water and get the pvc nice and warm then get it as tight as you can by hand. If you don't have any leaks, you're good to go.
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