i dont want to sound like a bad guy, but those intakes are no good, they draw hot air temp from the engine bay, causing power loss and faster overheating.

Are they compatible with the stock air boxes?

I would say yes they are, but don't quote me on that, it does seem to be installed out of the way.

I have catch cans on my turbo Veloster but not on the Z. Boost makes more blow by and crankcase contamination + direct injection and I have been shocked at how much the catch cans have collected between oil changes. That said, when I changed out my intake manifolds with the Z1 400HP ported manifolds, I was very and pleasantly surprised how clean the ports and intake vales were, quite pristine actually(32K miles). So I think they really are not needed on this car in NA configuration. Also, I have tracked this car so again, quite surprised how clean everything was. Will they hurt, no and they may help to some degree depending on the age of your engine and how you drive. Your mileage may vary.

mine are almost full after only a couple hundred miles

Considering the 370Z is port injected in both NA and FI, skip to 2:33 of this linked YouTube video. There are all of 10 seconds to describe an oil catch can for port injection, vs 10m56s for direct injection. Really just adding weight.
https://m.youtube.com/watch?v=T-B4VRxAtbw

In the esteemed words of Lil Dicky, “$ave Dat Money”

I installed the ADD-W1 dual catch can kit at 37k miles and at 39k the driver side can had a little over an ounce of oil goop that registered on the built in dipstick - not much but glad it didn’t get put back in my intake.

I've recently added this to my 2014, but the hood does not shut correctly. Off about a half a cm. Did you need to do any modifications to install? I have factory air boxes.

That EE guy is an embarrassment to engineers. He doesn't know what he's talking about half the time and mixes facts with outright ignorance to sound more intelligent then he actually is. He lost all creditability when he said that resting your hand on your shifter will damage your transmission. On port injection fuel does pass over the intake valves, it is no where near enough to wash and clean them from oil vapors from the crankcase. Nor is fuel going over important components like the throttle bodies that can gunk up over time.

Eh.....that gear shift thing does have so truth to do. On trucks with the G56 tranny. It's a know fact that if you rest your hand on the shifter. It pre-loads the synco's. Leading to wear in them. The shifter is so long, and goes straight into the top of the gear box. People don't realize how much weight they are putting on it. The one Dodge forum has a couple of threads on it. The shifter set-up in a Z, it wouldn't matter. If you did or didn't.