You may have a clog in front of your sink as the check vent is lower than the drain on the sink. Snake it or call a plumber.
Kitchen sink drains are a fertile environment for certain forms of life, and these can cause the drain to run slow. When hot water is run down the drain, it kills some of the lifeforms, allowing the drain to run freely for a while. Try using a drain cleaner, or a drain snake, to clean out the drain. Never pour grease down a drain, as it cools and solidifies quickly in the pipes.
It's possible that you have a blockage, on the toilet line and the water is coming up in the tub to alleviate pressure. Or, your vent may be clogged causing what is called back syphonage, causing the water not to push out.
There is usually some space around the drain hose where it goes into the drain. This usually allows enough air to get in that it is not a problem. If the washer hose is connected to the drain tight, then yes, there needs to be a vent above that connection. There may already be one above the drain if you follow it up. Most washer drains are fairly close to the main stack which allow for venting. The only question is, Does the washer drain alright? If so, don't worry about it.
Drain is partially clogged or the vent pipe is stopped up. Also check the vent holes underneath the rim of the toilet and clean them out.
Water, drain, and vent lines that are used to plumb a house.
You can use either a "Chicago Loop" to take care of the vent problem. Details can be seen here: http://en.wikipedia.org/wiki/Chicago_Loop_%28plumbing%29 Or another way is tio install an air admittance valve (available at Home Depot).
No.
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.
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.
The drain pipe should have a vent that goes up where the sink drain goes into it. That is where it vents. Sinks drain because there is nothing causing a vacuum when the water leaves the sink.
Heck no as the sink is higher then the basement drain and water does NOT flow up hill BUT it does seek its own level BUT you can connect a sink drain to a basement drain if you properly trap and vent the line
Take your snake on the roof and clean the drain from the vent.
The sink doesn't have an adequate vent.
Shouldn't be a problem. You may need to extend the pipe at the sink a bit to put in air vent so that the drain can vent. It is a checkvalve that lets in air when the sink drains. Or you can tie into the existing vent stack if possible.
You need to check with your local and state building codes, but I have done this many times before. Just make sure that it is utalizing a trap. If you go into the other drain below the other trap make sure you plumb in a new trap for the new sink