Am I bypassing the sound card when I plug the USB headphones in?
Yes, in order for you to actually get any benefit from the fancy sound card, you actually need to plug in a device into it.
Am I getting the surround sound from the processor in the headphones
or the motherboard?
Headphones work on 2.0, which left and right speaker. I don't understand what you mean by surround sound? as in a 5.1 setup. left, back left, right, back right, center? then you're not going to get surround sound period.
Would it be better to plug analog headphones into the sound card.
Yes of course, this way in fact you're actually using the sound card, and not the motherboard/usb interface. I don't understand why you have an impression that USB is better than 3.5mm audio jack. Most people would argue that analog is better than USB in every respect.
Can I use a USB to standard headphone jack adapter then run it through
my soundcard. (how well would that work?)
You could try. I've never personally tried this before, i don't know if it will even work. But if you're not willing to buy new headphones, it's worth a shot.
Is the sound card just a waste of money when it comes to headphones.
Pretty much these days it is. Because the built in audio that motherboards come with these days is will more than be enough for the average gamer or PC user. especially for the fact you're only using headphones.
What headphone and setup is going to yield me the best sound quality
for Battlefield 4?
This question is really mostly an opinion based question, can I really can't give you a certain answer, but I will say this, know what you're budget and spend wisely.