For cold weather gigs, I always bring warm boots, fingerless gloves and a wool hat. If you are wearing proper clothing for the cold then the next step is to make sure your hands, feet and head are warm and dry. That will make a huge difference in how you feel. Warm feet and hands are the key to outdoor gigs, as the capillaries in your hands / feet act as radiators. Keep them warm and you will keep your heat.
At gigs when I expect rain or sub-30 degree temperatures, I always ask for a pop-up tent for my gear, even if the stage is covered. This lets me make a game-time decision about my equipment; if it's really miserable and cold (or rain / snow are likely), then I lower the popup tent above my gear so that it's as low as possible without touching anything. Then I drape a plastic tarp around the side where the wind is blowing in. That keeps rain / snow out of my rig, and helps moderate the temperature a bit. Don't throw a tarp entirely over your rig unless it's powered down; it can overheat surprisingly quickly if all airflow is cut off. If you're on a deck against a building, you might try putting your gear as far back as possible and draping a tarp over the front of the rack and speaker cab. As
electronpirate notes above, you might want to bring a clear piece of plastic sheet to cover your MIDI controller.
Tubes should be fine in the cold. Once you warm up your amp on standby, your tubes may take a few extra seconds to reach operating temperature, but then you should ready to go. For the AxeFX, I can't vouch for the lowest recommended operating temperature, but Metallica used their AxeFX units in Antarctica...
Strings should be OK as well. I haven't experienced string breakage at lower temperatures, although it does make sense that they will be less flexible in the cold. I've noticed that when it's cold, it's pretty difficult to keep my guitar in tune because my hands contacting the strings will raise the temperature in that area, causing tuning / intonation to change. Once I move to a different part of the next, I'm warming up a different area while the previous area cools down rapidly. My approach is to tune between songs until we've done six or so; at that point it's about as stable as it is going to be, and I can just spot-tune as necessary.
The other thing I'd recommend is that once you finish your set, move your gear inside and leave it for at least an hour before you put stuff into cases or put on the rack lids. Your gear will be cold and once you bring it inside, you'll see condensation forming on it. You can't do much to avoid this, so just give your gear time to adjust to the warmer temperature and to dry out. When we're on tour and play cold or wet gigs, everything goes from the stage directly into the truck. We remove case lids / covers, close up the back of the truck, and run the dehumidifier inside it for at least an hour before we hit the road.
