A little background to answer questions before they are asked:

When I got my bike at the end of last summer it had a very slight fuel leak from one of the carb drain screws. The original tank had been replaced by the guy I bought it from because the original owner let it sit. I bought it in August 2014 with 4046 miles on it - it's a '97. Naturally, I planned for a carb cleaning because who knows if it was done since the tank was swapped. Inside, they looked pretty good. I replaced the leaky drain screw (I replaced both for good measure but only one was a problem). One of the floats was stuck but I didn't realize it at the time. The new drain screw stopped my leak...at the drain screw.

My bike now had the same magnitude (incredibly slight, barely noticeable) gas leak from the air box drain hose. I went back in and unstuck the float. Every post I found here on GSTwin said that was my problem. Unsurprisingly at this point, the leak remains.

I realized my major error, or rather my major miss. My leak was not isolated to when the bike was running. My vacuum petcock, as you all know, has On, Res and Pri. Since I've had the bike it's been on, On. I've never needed and therefore never used Res or Pri (yes I know what they are for). I now realized that my problem is the frame petcock. I reach my hand under and behind the lever on it and it's wet with fuel.

So now I arrive to the understanding my petcock needs to either be rebuilt or replaced. I'll be pulling it apart for inspection some time this week. My question follows: many people suggest replacing it with a non-vacuum petcock (I don't know what the technical term is). One with On, Res and Off. Where do I find one like this for my bike? Do I need a specific one? Does it still have four hose connections?

Side question semi-related: the breather hose (I think that's what it is) is missing from the tank. The port is there with nothing connected to it. Does this or can this cause any problems, related or unrelated to my current issue? Should I get a hose and connect it or is there no point?

Thank you all in advance. You've all been helpful in the past and I appreciate it.

Disclaimer: I did search and people had suggested certain threads for help in this matter but they don't seem to exist anymore.

My frame petcock was fine and I opened up just to look. Well I goofed it up and now it had a leak.  I searched and found there are NO rebuilt kits for it, the only option was replacing. So to Ebay I went and found one for like $40.  The other option is a bike junker (used parts). The new ones from the dealer are like $95.  I got new fuel lines too.  Now no leaks. 
The tank hose is just for over flow gas routing down and away from the bike. I was missing one for a while and found a hose section to fit and put it on.
Back to the frame petcocks. Some searches I read were referencing a different style petcock w/o the vacuum. We that relies on your properly working floats to stop the fuel flow.  Other options may remove the reserve 'tank' option. I have needed it a few times and was grateful to have it.

Hopefully this answers your questions;

Before replacing the petcock with a new one, make sure its the cause of the leak, not just a cracked fuel line on the input or output sides. On a bike more than 10 years old, I would replace all the fuel line first, rubber has a shelf life, and oddly enough tends to get more brittle the less its flexed.
If you isolate the leak to the petcock itself, I wouldnt expect to find a rebuild kit, as its kind of specific to the bike. I would look for a new or barely used one on ebay. Im pretty sure they used the same one up to the 09's so a low mileage petcock should be easy to come by.

As far as the tank vent you are referring to, that is just an overflow line. Our bike's tanks are really a tank within a tank, and that line is there so when you overflow it at the pump (or due to expansion on a really hot day with a full tank) the fuel has a safe drain point. This is likely not the cause of your leak, but its worth looking at to make sure you arent mistaking a petcock leak for some overflow at the pump station.
I would strongly recommend atleast a small length of fuel line from that point to somewhere below the motor, just incase of an overflow, your not pouring raw gas onto a glowing hot motor.

And when it comes to replacing the vacuum type petcock with an on/off/reserve function, there is a great write up on this in the "racers corner" section of the gstwins site. Its not as convenient as the frame mount, but as far as I know, its the only easy way. I havnt seen any members with a frame mount non vacuum petcock, but that doesnt mean it doesnt exist. In fact, any decent machinist could make one at a small cost. But from a price point, you will be better off with the stock unit IMHO