No! unless you want to always have problems with your sink. The sink (if its higher then the main line) will drain for a while but it will plug up sooner or later because the water will equal out till the dirt and junk fills up. You always want to use common sence and remember the water loves to go down hill!
Yes, i would advise you too.
Yes a clogged VENT pipe can cause a sluggish drain
a 3 inch pipe
The pipe out of the wall usually only sticks out a couple of inches. You add what is needed to the sink drain.
Mission CAP or IPS plug if there is a female fitting
There should be a nut around the chrome pipe holding it to the brass. Loosen the nut and pull the chrome pipe out.
1-1/4" is typically the diameter of a bathroom lavatory drain pipe. 1-1/2" is more commonly the drain pipe size in kitchen sinks.
By checking the vent to make sure its working and that no one installed an AAV
That would depend on the height of the drain and how deep the sink is. As long as the drain is lower than the bottom of the sink, it should be fine.
A buzzing sound could be created from debris in the drain pipe. Try removing the P trap and clearing out any junk you find.
If it's a sink drain , best to remove the entire pipe and P trap. I then take outside and hose them out.
Radio Shack "drain pipe tracer".
Probably leaky drain pipe form toilet which passes through that room. Work out which corner of that room is below the toilet, you will probably see a swelling in the wall where the drain pipe is hidden. Call a plumber and get him to fix it.