while true that usb devices can be susceptible to usb noise, its also true that external devices can have less noise than internals in some cases. this is why external dacs are popular. having an external amp would be nice (perhaps one with both optical and 3.5mm/rca outputs) since it would allow use with your phone (if you bought a portable model) or allow a volume knob you can turn with your fingers at minimum. strictly speaking a soundcard like a Z or STX is still an option if you like that option better.
for strictly stereo audio.. music and similar i would say an external amplifier plus separate external dac... or perhaps a combination dac+amp unit would be a stellar choice.
if you're worried about usb noise, you could get a soundcard and export via optical to an amplifier or as an alternative use low-level (unamplified or low level amplified) 3.5mm outputs. optical would put a break in the line to eliminate any ground loop buzzing in case 3.5mm is how you want to connect. ground loop is an issue that some people experience but that many people do not (depends on how you have your devices plugged in to your house and how your outlets are wired)
if you have a chance to test out the headphones in person that would be GREAT and is far better than us just telling you what to buy.
if it is a proper store they will have an amplifier connected to the headphones after any 3.5mm cable you have access to (is this the case or does the cable go right into the headphones?). if they are not amplified and you plug into the headphones directly then the headphones will sound better with an amp than they do on your phone. if they already have an amp in the store then they should sound similar.
your budget for both headphones and amplifiers is perfectly fine and you have many choices. in fact, your budget is even higher than you need (depending on your choices of course as cans like the hd600 can start to eat up budget quick).
try the dt990, dt880, dt770, hd598, hd600 in store if possible and let us know what you think about how they sound. this would be helpful.
remember... your choice of amplifier can affect the sound. if you used a clean sounding amplifier like the schiit magni (used often in a dac+amp external stack modi + magni) something like the dt990 then they will retain the harshness and cool nature to the highs while if you used an amplifier like the fiios (which are classified as warm-sounding) or tube amp (which generally have a warm signature... but also have a very particular way audio sounds out of them) which would bring down the harsh edge on bright treble and push the entire signature a little more towards bass tones. you can do similar with EQ of course, but amps have a somewhat different sound effect than EQ.
let us know your thoughts on the headphones above when you've heard them in person (or the models you can find and listen to in person) as well as your preferences on if mobile is necessary and what your thoughts are on the possible powering configurations we listed.