Bit rate tells surprisingly little about quality in the case of mp3 files. There are old encoders where no matter how low you set the compression, there will always be audible artifacts. But LAME and the like have long gotten over this, and properly done 320 kBit mp3 is for listening purposes lossless, just like CDs are. Note anyway that mp3 is an outdated standard; I'd like to recommend switching to OGG Vorbis but practically this wouldn't gain you anything.
What digital connection you use to transfer the file from the source to the DA unit has no influence on the sound whatsoever1. WiFi is fine as long as its avarage transmission rate can keep up, which should never be a problem nowadays (apart from complete dropouts if the connection breaks down somehow, of course).
Laptop on-board sound cards are horrible, so off the top I'd say that's your bottleneck. However, the one thing they usually manage quite well is driving headphones. The main problems that such sound cards have are
- Awful mic preamps.
- Bad AD/DA units.
- Insufficient (or no) shielding against interference from power supply bursts etc.
You don't need the mic inputs. Even low-quality DAs of today can actually reproduce properly bandlimited audio quite faithfully. And the interference issues mostly turn up when you connect to some other audio device (with ground connection), but not so much with headphones. The headphone amplifiers themselves aren't great of course, but since that's quite an easy job electronically they still don't have easily audible influence on the sound. One remaing problem is high volume levels, especially in mobile devices, for power-supply (as well as, possibly, heat dissipation) reasons.
You may try an external sound card, it might improve you listening experience; but perhaps not significantly.
Like any kind of speaker device, those actually have more influence on the sound than any properly designed purely electronic component2. That's often not so much considered a flaw as a characteristic; HiFi speakers aren't really designed to transmit with flat response at all, but to transmit "nicely sounding", which obviously isn't quite well-defined.
Studio monitors and -headphones are, in principle, designed to operate as linearly as possible. Incidentally, this means they don't really sound "good": they're most of all supposed to reveal any issues in the material. Sound engineers value this, obviously. But again, mastering engineers anticipate that most listeners will use technically inferior speakers, and set up the mastering accordingly.
This, in a way, means the actual bottleneck is probably the orginal source; not the file format but the mastering. Indeed many of today's records are mastered quite horribly IMO, grossly overcompressed etc..
Of course this depends on what style of music you hear. Classical music and Jazz are least affected, Pop through Electronic and Metal are most. What can you do about it? It's nothing you could fix by buying better listening equipment; if anything you could get worse equipment to better match the master engineers' expectations.
Just try and be selective about what tracks you listen to, that's the advice I'd give you. A great recording3 over 30$ equipment is much more enjoyable than a bad recording over 1000$ equipment.
Actually, that's only true for asynchronous transport, as in your case. If you need realtime performance, then there are sure enough a couple of issues in the digital transmission.
2Of course, I'm leaving out deviced designed to alter the sound here, like EQs etc..
3Sure enough, in the recording studio they will have needed expensive equipment. Recording has much higher demands (e.g. low-noise mic preamps) than listening.