Yup, no lag at all    actually i didn't use all the apps, the apps i use the most is candy crush saga, wonder zoo, temple run 2, granny smith (really nice running game), ski safari, my little pony, talking angela and many other else..

and messaging apps and some reading news app..

Hardware limitation, I think all phones should at least come with a headphone amp these days.
Majority still uses earbuds, but there are plenty who are willing to switch to full-sized audiophile headphones if their equipment can cope along with it.

Well, Power Amp does have an EQ so you can sort of "fake" the gain a little bit by adjusting the Preamp.
Though that wouldn't help much at all because the limitation of the S4 is that it simply just doesn't supply enough current to 50ohms headphone.

I was testing it with a Sennheiser HD558, which are known to be very easy to drive but fluctuates a lot. I get distortion on low frequencies, which I didn't quite like either due to the high output impedence on the S4.

I have Power Amp and Winamp as my main usage for music players. The One's stock player is pretty decent for what it is, just lacking the option to adjust EQ. Otherwise it is fine for what it is if you want all your music to be flat.

As for the Note 2, I do have one lying around but haven't tested with it yet on anything higher than the standard 32~36ohm headphones. But the Note 2 does indeed come with a Wolfson DAC (pretty good quality DAC btw) and will complement the Phonaks 232 pretty well assuming it can drive them — which I think they should.

Also check this out if you want to use the Note 2 as a music device - http://forum.xda-developers.com/showthread.php?t=2039716

Nope, beats doesn't work with anything else other than the stock player. It's tied to it.
Turning on the beats EQ doesn't change the sound signature from PowerAmp at all.

Again though, I hate the beats EQ due to how they try and over-emphasize on bass. It's so muddy I'd rather drown my ears with wax. It also boosts the preamp and this gives the illusion to users where their earbuds/headphones would sound "louder".