If I lay a Bag of Holding (BoH) on its side, and have a hose half in and half out of the BoH, will air flow freely through it?
Or will the opening of the BoH block the air from flowing?
There's nothing in the description for the item that describes or even implies whether such a thing would work. Nothing in the item description gives us any idea of how the bag opening works mechanically or even if air flows freely into it when open.
The only description that implies air in the bag is:
Breathing creatures inside the bag can survive up to a number of minutes equal to 10 divided by the number of creatures (minimum 1 minute), after which time they begin to suffocate.
The implication here is that there is a finite amount of air in the bag, but nowhere is it described how air gets in or even if it can. A strict RAW reading would note that nowhere does it allow for more air nor does it strictly even say that air is the issue.
Also covered nowhere in the description is how something would act if it sticks partially out of the bag.
Such ambiguities are thus solely left to the discretion of your local DM to decide for their table.
From a scientific perspective, the opening to a Bag of Holding, and presumably the haversack and portable hole, is an air-tight door.
So when you "seal" the bag, it is not that the air inside escapes. Otherwise you'd have a popping sound every time you opened the bag as you break the seal and all new air rushes inside.
When a living being is inside the bag, they don't run out of "air", they run out of oxygen. Living beings as we know them breathe in oxygen and breathe out carbon dioxide. Plant life absorbs carbon dioxide and release oxygen. We are symbiotic that way. But you couldn't just throw a fern into the bag and think it would supply you with oxygen. too little, too slow.
So the hose you are asking about would work as a snorkel; allowing new oxygen to come in and expel carbon dioxide. More science would be needed to figure out how big the hose would have to be to circulate the volume inside the bag.
So the real question is, can the hose stay part way in and part way out? That also is up to the DM, but this question also indicates that there is no clear cut answer in the rules.
My personal ruling would be that so long as the bag is not moved, then the hose is fine.
In your case, if someone tried moving the bag more than a few feet, the hose would likely be ejected, unless something inside was holding it. In which case a DC20 Strength check would let it be pulled all the way inside (assume there was volume left to hold the whole hose).