Just a home-gamer, not a contractor here:

The p-traps i have replaced have not needed putty; they have all been fine with just the slip-on ring seals and screwing them down.

It is good to note: make sure they are on the right way (stupid younger me talking)

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Some traps are intended to be glued. Sounds like OP’s is a screw together kit that doesn’t need anything but the included ring seals.

Correct.

Many Thanks

No. Just wrist tight, run some water and check back in an hour, you want zero drip

Rubber gaskets in plumbing joints don't need or want anything else. Only use plumber's putty when setting a drain in a sink etc., only use pipe thread tape on tapered pipe joints. Compression fittings are assembled dry. Plumbers guffaw when they see sealant used where it shouldn't be.

If the new P-trap is plastic, you don't need putty, and absolutely don't use pvc cement unless you want to replace it every time you need to clean the trap!
The seals should be fine. The only place you should be concerned is where it transitions into the main waste. The transition fitting should be cemented to the waste pipe, with the proper seals there. If its a metal nut to a metal transition, make sure its clean and the seal is positive on that pipe (that's the only place I've ever used putty - transitioning a new pvc p trap to an ancient pipe on a house built in the 1930's)

I always put plumbers putty in the p-trap threads gives me more sense of security and won’t leak. Takes an extra 30 secs for peace of mind. The ring seals especially rubber will eventually rot out but the putty will stay there.

Do you put plumbers putty on rain downspouts, or roofing tiles? Plumbing traps are designed to permit water to flow past...but whether they're sealed during a clog you'll have to consider. Drains don't clog much...and when they do there's not often much static head of water weight - provided all orings or gaskets are in good condition that's all you should need...but I cannot see your trap/pipes - you say pvc cement so they're white/grey plastic...or possibly abs...you should absolutely NOT cement them...as that defeats the purpose of being able to easily disassemble. The rubber gaskets should maintain the seal provided your alignments are good. The goal is to get the gaskets to do the proper sealing.

You can run hot water through the pvc ptrap and then snug the nuts at bit more. You don’t need to crank on them, just enough so they don’t drip, and then maybe a quarter turn beyond that.

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