Are IV's of much importance in PvP?

It interesting thing is they dont really matter (unless its a mirror,like giratina v giratina, say ones a perfect iv one, but lower level and the others a 0/0/0 at a higher level with the same CP). Generally its movesets that are much, much more important, then CP. But there are some iv spreads that get pokemon to higher levels at the same CP as the same pokemon with more IV's

For pokemon who's Max CP is bellow the CP cap, so think medicham for great league or blastoise in ultra, the higher the IV' the better, but for something like giratina in ultra, iv's dont matter a whole lot.

They can make a difference, albeit a fairly small one. In the great and ultra leagues (with the exception of Mons who max out under 1500/2500 CP), a Mon with 0/0/0 IVs can perform roughly the same as the same Mon with 15/15/15 IVs. The only difference is that the 0 IV Mon will need many more power-ups to reach the same CP range as the hundo.

I agree with what Metagrossmaxis stated in that IVs aren't as important as other factors when choosing Mons for your PvP team. In my opinion, I'd say movesets > typing > base stat balance > IVs as far as order of importance goes.

Also, if you have an Android phone, I'd highly recommend downloading Calcy IV (it's one of the safe apps). It can help you calculate how close your Mon can get to hitting the 1500/2500 CP benchmarks and at what levels. It will also let you look at what that Mon's attack/defense/HP stats will look like at those levels.

Very rarely. Close matchups, like exact mirrors, may sometimes be decided by them, but it's more likely that clock sync or the bug with who gets the charge move off first is the deciding factor.

The exception is mons that max out below 1500 or 2500 CP. You want to get as close to the cap as possible and if your say, Medicham, has bad IVs, it might max out at 13xx instead of 14xx, which lowers its stat product and therefore TDO significantly and may make it lose consistently in matchups that a perfect Medicham would win.

tl;dr not that important for most mons if they can reach the CP cap.

Gyms and raids are a race against the clock with no CP limit, so higher IVs are better with an emphasis on attack IV.

IVs matter in PVP, but in a different way. In some ways, IVs matter even more because you can't just generically say "bigger is better."

PVP is an endurance fight with a CP cap. Attack is weighted in the CP formula, so cutting down a little on attack allows more total stats to be added in defense and stamina. Also, Price is Right rules, you want to get as close to the CP limit without going over. A few IV points could be the difference between a sub-1500 CP and going bust with 1501+ CP.