1- the ad500x/ad700x are widely respected for their vast soundstage and good positional audio. they are really quite good for things like fps gaming. the hd518/558/598 fall into the better than average category. more than good enough, but not quite excelling.
2. depends on what you expect sound signature wise. both are good in their own right but what fits you may vary. i'll cover this after.
3A. if you like the headband the ad500x/700x do have comfortable earcups. the headband is a love or hate scenario. if you like it you'll think they are comfortable while if you hate it you might dislike the headphones because of it. hard to say which unless you try it. hd hd518/558/598 lineup is in the average to better than average category.
3B. second part to this is sonic signature. the hd*** series is really laid back and not quite as detailed nor bright on treble so its a good easy listening headphone for hours on end. while it might be a bit less exciting than some cans it depends on what you're looking for. some people love the mellowness. the ad*** series is more treble/mid focused and is brighter and more detailed all around. while this can be good for the added detail and vibrancy in those ranges, some people can get fatigued by treble.
4. for the cans listed above, not normally unless your onboard is extremely poor or if you want support for virtual yet lack it. for some other choices (below) you might need an amp or soundcard.
5. i'm getting to that..
schiit fulla... you may laugh at the name but schiit is an extremely well respected company. the fulla might work for some easier to drive headphones but for those a soundcard may be the better option or the fiio e10k. for larger cans, its worth looking at the schiit magni+modi, or just the magni, the audioengine d1 or possible the o2 amp + odac. it really depends on what headphones you get. for the ones above, they are easy to drive.
$300 total counting amplifier or soundcard?
as for alternatives or what is best for you.. that depends on what kind of sound signature you want and what properties you want in a headphone.
-sound signature?
treb/mid focus - good for picking up details in fps games and not having things like footsteps covered up by bass. may sound a bit anemic to bassheads and are a bit less fun sounding due to lack of bass for some. (ad500x/ad700x)
flat/neutral - good for general use and more faithful reproduction of sound. lack of heavy bass which can cover up things like footsteps so good for gaming but not at the level of bass light cans. enough treble energy but not to the level of treb/mid focused cans. some may find flat cans boring. (hd518/558/598)
v-shaped - more bass and treble with recessed mids. good treble energy for picking up small details while having bass for things like explosions. generally considered one of the most fun to listen to sound profiles by many. bass may cover up some treble like light footsteps so they are not as advantageous in fps gaming but certainly can be used if you want a can that is both exciting and decent for gaming.
bass heavy - bass cannons in other words. generally bloated and extreme bass levels which means less treble and early mid levels. tend to be a little distorted sounding due to bass focus. good for bass heavy music or bassheads.
open vs closed?
generally open is considered the most ideal for gaming due to wider soundstages (depth or distance from center point to where sound appear to come from when listening) but leak sound in and out. they tend to be lacking in bass or at the least not bassy. closed cans lack a large soundstage generally but do isolate you from noise in and out and tend to have the capability of more bass (capability to.. this does not mean closed always means more bass). these are just relative and of course there are exceptions to this rule
other options?
if you want something in treb/mid focused realm some cans from akg would be good. k701/k702/q701/k7xx. they tend to be power hungry so you would need a good headphone amp like the magnii. they tend to be picky about sources.. on good quality equipment they will sound GREAT but they will really pick apart things like poor quality audio files, bad radio streams or crappy equipment. the closed version of the ad500x/ad700x (a500x/a700x) are okay if you need closed cans but the open variants are better due to soundstage.
if you want something neutral, the dt880 from beyerdynamics perhaps. lacks the bass of the dt770 closed cans and doesnt quite have the treble brightness the dt990 has making it a little more flat and neutral than those two. a good choice under $200. i would highly suggest an amplifier but you might be able to get by with a creative z.
if you want something v shaped, the dt990 is an excellent value can. very bright crisp treble and for open cans surprisingly good bass output. well known comfort (just like the dt770/dt880). i would suggest an amplifier. the dt770 is a closed can with a more bassy sound but works well for people who want something with a bigger than average soundstage for closed cans. the fidelo x2 also falls into this category and if you want something with a bit less treble energy than a dt990 and a bit less quantity of bass but somewhat deeper frequency response they might be worth looking at.
also of note for headsets. most are rubbish but a few are based on actual headphone designs. those i know of are listed below. generally it makes sense to go with the headphone variant instead (normally cheaper) but some people absolutely want an all in one..
beyerdynamics mmx300 is basically an overpriced not quite as bassy older variant of the dt770 premium.
sennheiser game one is basically a hd518/hd518. prices last i looked were fairly comparable.
sennheiser game zero is similar to the pc363d, pc350se, hd555 and hd595
audiotechnica ad1x is basically an overpriced ad500x/700x. ag1x is the a500x/a700x
hyper x cloud is based on the takstar pro 80 headphone