The vent for the drain is in the wall. Where the drain goes in, it T's with the water going down and the vent going through the roof. With a double sink, one side is the vent for the other. As long as the water isn't pulling against a vacuum, it will drain.
Yes you have to have one
If there is no vent off the kitchen sink you would need to install an in-line vent.
yes.
It is usually hooked up to your kitchen sink p-trap or garburator under the sink, which is vented. This is the only vent you need for drainage.
If the drain is used as a wet vent for another fixture 2". Otherwise 1.5".
Take your snake on the roof and clean the drain from the vent.
There is a vent available for this. It is meant to cap 1 1/2 PVC. T the drain and put the vent as high as you can. It is spring loaded and the water will pull the vent open and close when the pressure equalizes. This is only for a single sink. If using a double sink, one side uses the other for the vent. You can use the add on with a double, you just don't have to.
By properly sizing the vent by fixture units and developed lengh and pitching it up from the fixture it serves
Partial clog or not enough or proper venting. If it is a single sink, there may not be a vent to it. On a double sink, one side vents from the other. On a single the water is trying to push past the air in the drain.
vent in bathroom manditory
You have a blockage in your main sewerline. You have a problem with your venting system. This is why some codes now require a vent within 6" of the toilet. If you will add a vvent to your main line at the toilet you will stop this. Check to make sure your vents are not clogged by a bird or bees nest first.
A vacuum is being created at some point after the two drainage pipes meet. This forces air back up the other pipe and through the "U"-bend underneath the sink, causing the water to gush upwards and drain again, making the gurgling sound. Since both the bathroom and kitchen gray water supplies contain no sewage, this not harmful.
Make sure the drain line and the vent are cleared. Noises like that are usually caused by either restricted flow or a plugged drain vent.