The next thing to check is what kind of cables you're using. You need a return path for things sent from the Axe to show up on the LF, and the fact that you're not getting tuner or tap tempo data coming back in but do have those enabled on the Axe makes me think you don't have a return path working. You need to use one of the following:
1. A single 5 pin cable, with all 5 pins wired. Note that while just about all MIDI cables will have 5 pins, they won't always all be wired. If you have a multimeter you can check. You should not use a second MIDI cable for Axe MIDI out -> LF MIDI IN in this case.
2. A single 7 pin cable, with all 7 pins wired, if you're sending phantom power through. Note that it's been reported that the rocktron 7 pin cable is not a true 7 pin cable. It has phantom and one direction of MIDI, but no return path. Again, with a proper 7 pin cable you wouldn't use a second cable. If you have the rocktron cable already, you might be able to use a second cable to work around its issues, but only if it's the rocktron cable.
3. A pair of 3 pin cables. When you're using a pair of cables, they *need* to be three pin. If they're wired on all 5 pins, even if they're a combined dual-head cable, you do NOT want to use the Axe Out -> LF IN connection, as that will actually create two return paths and confuse the unit.
Best solution, buy this for phantom, custom made in the length of your choice:
Best-Tronics Mfg., Inc.: MIDI7-XX
OR, if you don't want to use phantom, this is their 5 pin version. Though 5 pin versions can be found more easily (and perhaps cheaply). But best-tronics makes good stuff. I would highly suggest them for the 7 pin -- theirs is wired properly and well made. Phantom is very convenient (though you'll need an adapter to plug the LF adapter into the Axe, if you didn't order that when you got your LF). But if you don't want to do phantom, here's their 5 pin.
Best-Tronics Mfg., Inc.: MIDI1-XX
If you're confident you have the right cabling, and that it's working properly, I'm not sure what else the issue would be. But your symptoms do sound like they'd either be the SYSEX setting on the Axe, which you've checked. or your cabling.