Is the bike stock? Meaning stock pipes? You've already verified the airbox is in place so thats why Im leaning towards stock and never rejetted. At the very least this rules out the top caps of the carbs and a pinched/damaged diaphragm.
You don't have to remove the carbs to get to that fuel filter but its a pain. I haven't had an air box in either bike for such a long time I don't even remember how to get to it if the air box is there. Sorry, wish I had better info here. You may want to put this on the back burner for now tho.
Can you take a pic of the cracks in the air box outlets and post them so we can see how bad they may be.
I make the above statement because I saw something in your second pic that triggered another option for you to look at but first that hose you asked about... its ok. The air box will have a drain hose or two and the fuel tank will have a vent hose. All of those come out under the crankcase. The vent hoses from the air box may also be plugged and if you find that its ok and normal.
Now... the other thing I noticed is that your bike has all the Ca emissions crap on it. You'll have one or two solenoids and a couple carbon canisters (one is in your second pic next to the regulator/rectifier). As I mentioned earlier I have two bikes in the garage, both 07 Americas. One had the Ca Evap stuff (Karens bike), the other did not (mine). When Karen had some problems a couple years ago the symptoms were poor running, low power while twisting the throttle and dying or not idling as well as some very hard starting when the issues were present. It would happen whenever it felt like but would clear up until it happened again. I removed all that emissions stuff and have not had an issue since. Below are links to the Bike Bandit site so you can get a visual on what Im talking about. Seems there were two systems used (I had the first one).
The solution on Karens bike was to remove it. Since we live in Wa I had no issue removing all that junk. When I did it left a couple vacuum ports open on the carbs but that was a simple fix. I used pieces of vacuum hose to link them left and right (not front to back). You can also go get some vac caps at the local auto parts store and use those to plug them. If you do this make sure the caps are snug and not loose in any way. Vacuum leaks will screw you us as well.
http://www.bikebandit.com/2002-triumph-a...4727-and-higher http://www.bikebandit.com/2002-triumph-a...54726-and-belowI would suggest getting some vacuums caps (squirrel condoms) and bypassing the current hoses that are there. You don't have to remove everything, just the evap system hoses.
Caution...
You probably also have the AI system still installed. The most common tell is the silver/chrome tubes behind the spark plugs. This is also part of the emissions control and there should be one vacuum tube going form carb to under the fuel tank. You should leave that hose in place. Heres a link to a pic of that system:
http://www.bikebandit.com/2002-triumph-america-emissions-controlWhile your checking all this stuff check your plug wire connections at the coils. Under the tank, towards the front you have two coils (one for each plug). Make sure the wires are in there and are not too lose. You should not be able to pull on them and have them just fall of. There should be some snugness to the fit (sorry, my big boy words are escaping me right now).
