Which to keep?

My guess is that defense and stamina will be a little more important in PvP than attack, simply because the CP formula weights them less heavily than attack, so you can buy more performance in battle with the CP 'spent' on defense and stamina as opposed to the CP 'spent' on attack.

That said, balance will be the name of the game. Tanks won't be optimal, nor glass cannons. Just where the right balance point will fall is an open question.

That said, stats aren't simply "biggest rock is best rock" anymore. Save what you want, or maybe highest level.

For collection mons I only keep highest IV period. If you have no intention of powering up or using a given mon, it's IV will never be a factor.

I would (and do) favor the highest IV total for mons that are kept purely for collection value, there is no reason to favor attack stat over the total if you never intend to use it as an attacker.

Since this question generally answered, how have I seen 59 and caught 48 Spinda? I have no research pokemon stacked, to my knowledge never lost a stack of research pokemon either.

If it's something I never plan to fight with I'll go with the highest overall IV and call it a day. A L10 perfect Pidgey warrants more bragging rights than a L35 mid-IV Pidgey.

If it's entirely useless, IVs.
If it's potentially usable in a unique 6 or draft challenge some day, level. Not that I see myself every doing those challenges - I just see no purpose in holding a 98% iv version of a useless Pokemon .

A 100% Pokemon may be worth keeping for collection value if you care.

Otherwise I never care about IVs at all unless I intend to power it up. PvP may change that slightly (community weekend was awful trying to iv check every single bulbasaur and others to see who gets closest to 1500/2500 without going over).