Clogged drain line downstream of basement sink or possibly problem with venting.
Additional shut off valve, below sink or in basement. Incomplete pipes not tied into bathroom sink line.
Should be individually vented
Yes, it is done all the time
No, stand pipe for washing machine is meant only for washing machine. Tie sink into drain line that washing machine stand pipe is tied into. 2" minimum drain line should be what you tie into. Sink drain line use 1 1/2".
yes.
That would depend on where the floor drain was in relation to the sink. If it is the drain for an existing sink it may be too far out from the wall. There is no way to know without measuring. The pedestal may have a hole in the bottom of it that may be big enough for the drain or not. Depends on the manufacturer. If you can get it to line up somehow, there is a trap that drains out the floor instead of the wall.
Most likely a drain has dried out and there is no water in the trap to stop the gas from coming up. Sink, shower, or main floor drain.
The most nessecary tools needed to fix a clogged sink are a plunger to extract the material clogging the sink and a sewer snake to hook and extract the debris.
Ask This Old House - 2002 Splitting Logs into Firewood All About Paintbrushes Installing a Drain Pump for a Basement Sink 9-23 was released on: USA: 12 March 2011
sounds like you are going to need to install a sump for the sink to drain into and then you put a sump pump at the bottom and connect it into the houses main line.
The trap in the drain line under the sink is dry. It should have water in it to stop the smell