All in-line wired remotes designed for iDevice will work with Note 3 to play/pause/call, double-click to change the track, and long press to start Google NOW. You can use built-in mic to make calls. The only thing that will not work is volume up/down buttons. Those are specific to apple. The only company that so far announced official Samsung compatible headset is Sennheiser, but these headphones are not available yet. I will have a review sample right before the release to share with everybody.
You can check out my massive list where I have reviewed hundreds of different accessories, including dozens of headphones:
http://androidforums.com/android-accessories/796475-twister-reviews-android-accessories-gadgets.html
If you want full control, you need to go wireless for a full compatibility with your phone. BlueBuds X are OK, but overpriced, and I would highly recommend BackBeat Go2 - much better sound quality, a lot cheaper, and comes with a charger case.
For wired headphones in your price range, Kiipsch S4A will work, excellent sound quality. Logitech/Ultimate Ears UE600 is excellent, but they are refreshing their entire product line so wait until the new models will be revealed soon. If you want to dip your feet into budget audiophile world, Shure SE215 is an excellent choice and it comes with removable/replaceable cables. A few ebay vendors make replacement cables with full Samsung Galaxy remote control including volume (I reviewed those cables in my UE900 write-up). MEElectronics A151P with in-line remote will blow you away with sound quality and available on Amazon for $50 now:
Amazon.com : MEElectronics A151P Balanced Armature In-Ear Headphone with Inline Microphone and Remote (Black) : Electronics . But as you know, YMMV since sound quality is very objective thing. I review a lot of audiophile quality headphones in $400-$500 range, but at the same was very impressed by $9 set of AWEI headphones I reviewed here:
http://androidforums.com/android-accessories/765018-review-awei-in-ear-headphones-w-lots-pics.html
You have a lot of choices, and it can get overwhelmed. Perhaps the best thing is to go to your local Best Buy or Staples and listen to some of the headphones on the display in there? Otherwise, take your chance and order it on-line. With amazon it always hassle free returns
Good luck!