First of all, it looks like my fuel tank has a separate "breather" hose separate from any kind of "EVAP" system. I can see it coming out from the bottom of the vehicle, and I'm guessing it probably connects to some compartment somewhere inside the top of the fuel tank (i.e., somewhere at least above the fuel level).
I'm thinking this is probably an overflow tube which is connected to the filler neck area somehow. You don't want gas flowing down over the tank and landing directly onto a hot engine. Could cause issues. If that's what it is, it actually would be separate from the tank and would not cause air to enter it when the gas cap is on and sealed.
My understanding of EVAP is that the EVAP system is responsible for all of the air coming into the fuel tank (to displace gasoline, that is). In that case, is the separate fuel breather hose simply there to help take some load off of the EVAP system? I thought that EVAP vent valve is always generally open, so it would always be the natural "breather" for the fuel tank anyway.
That is the main purpose of the EVAP system on a car (to provide venting to the tank) and don't doubt it works the same for your bike. As long as the first portion of this answer is correct, the rest of this portion of the question is moot.
Also, my understanding is that, during engine operation, all EVAP valves are open to allow the engine to consume the gasoline vapor from the canister. But, is the outside air coming from the EVAP vent valve at least somewhat filtered? Perhaps, is the EVAP vent solenoid actually forked off of the air-filtered portion of an engine's intake pipe? (That is, to ensure only filtered air makes it into EVAP)
On modern fuel injected vehicles, the fuel system is closed. As the engine consumes gasoline, there still needs to be some sort of balance to occur or a large amount of vacuum would happen in the gas tank. If too much vacuum were to occur, it would collapse the fuel tank. The only way to relieve this vacuum is to allow air to get back in. The EVAP system thus works two ways. It bleeds off any excess fuel vapors from the tank and it allows air to breathe back into the system to ensure too much vacuum will not occur.
As far as the cleanliness of the vented air, I would assume it depends on how the system is setup, but for the most part it should be. Remember, though, the amount of air it takes to put into the tank is just enough to replace the gas which is used. A gallon of air (at atmospheric pressure) is not all that much air. In an enclosed space where the air is not cycled or circulated, it should not really be a big deal, filtered or unfiltered.
Also, my understanding is that the vent solenoid is generally always open, and only closed during fuel system leak testing. Now testing aside, couldn't we just make all of the intake go through EVAP anyway? (After the air filter, that is!)
The EVAP system only purges gas vapors as needed, then only after the engine is warmed up. Remember, the primary purpose of the EVAP system is to provide a sealed fuel system and to relieve over/under pressure in the fuel tank. It will only absorb excess fuel vapor. Excess fuel vapor is only collected when there's an over pressure situation in the tank. If there is no over pressure, there's really not a need to collect excess vapor (because there is none).
Part of the EVAP system also checks the integrity of the fuel system, to ensure there's no leaks (inbound or outbound). It can do this by either creating pressure in the tank or allowing vacuum to occur. If the check fails, then you get the EVAP code. If the system utilizes pressure instead of vacuum to do its checks, I'd assume there's not going to be a lot of excess fuel vapor in the first place (reality could be different, as I've said, this is an assumption on my part).
Granted, I've also heard that EVAP is actually "closed" unless the engine is well above idle, or else it may disrupt the "air fuel mixture or ratio". But why would it disrupt it? My bike has electronic fuel injection, so couldn't the on board electronics simply adjust the amount injected from the main fuel line to balance out whatever you get from EVAP anyway?
You are correct it is a closed (or sealed) system. That's how it's designed. When it does purge (at least in pollution controlled electronic fuel injected vehicles), there is compensation for it. The EVAP system itself compensates by allowing air to go into the intake system along with the fuel vapors.
If you want to know more about EVAP systems, I found these two articles which seem to explain how they work in pretty good detail: