Great Open Headphones (Help)

you will want to note what kind of equipment you are using and if a suitable amp/dac/soundcard needs to be in the budget. also you should mention if you are recording (youtube streaming, etc) and if you're using an audio interface or similar. country of origin (guessing usa) should also be specified. if you want/need virtual surround please list that as well.

would you say your onboard audio quality (the dac) is good? or would you be interested in an external dac such as using the modi+magni stack (or o2+odac, similar)

it would be worth looking at the hd600, hd650, he-400i for good middle priced options or the dt880, dt990, k7xx for lower priced options.

i recommend reading some user reviews
http/www.head-fi.org/products/sennheiser-hd-600
http/www.head-fi.org/products/sennheiser-hd-650-headphones
http/www.head-fi.org/products/hifiman-he-400i
http/www.head-fi.org/products/beyerdynamic-dt-880-pro-250-ohm-semi-open-reference-headphone
http/www.head-fi.org/products/beyerdynamic-dt-990-pro-headphones
http/www.head-fi.org/products/akg-k7xx

on a budget if you want soundstage, bass and bright treble the dt990 has it. they can be a little bright so you might want to eq them down. the 880 is more neutral with less of a v-shape and less treble/bass spike than the 990 has. i dont know much about the new k7xx but from what i've read its a bit bassier and easier to drive than some other akg cans so it may be suitable for you.

the hd600/hd650 is a natural upgrade from the hd598 series with a good soundstage and is more laid back and easy listening than some other headphones. the he400i planar is good all around but does not have a huge soundstage.

there are of course other options, perhaps some of the others could make a few other recommendations.

i believe the he-400i has more bass presence than the he-500 while the he-500 is more neutral and less colored sounding. not too familiar with the 560. which you like best depends on what you're looking for in a can and how much you want to spend.

as for how headphones sound, again, it depends on what you are looking for. some are very forward with mids and treble but lack bass, some are fairly neutral throughout, some are v-shaped with subdued mids but forward bass and treble, some push mids forward and subdue treble and bass while others still are more focused on mids and bass. everyone has their own idea of what sounds best although flat or neutral headphones are often a jack-of-all-trades-master-of-none so get beat out in trebles, mids or highs by headphones which specialize in that kind of sound.

there is no perfect headphone which is great for every type of music, game or sound.

it looks like the audio onboard your motherboard has is fair. better than some but not the best found on motherboards. its likely usable as the dac portion although it would be a lie if i said an external dac or soundcard wouldnt be an improvement upon that.