Yes a clogged VENT pipe can cause a sluggish drain
Kitchen sink and washer use same drain line. Drain line clogged causing backup of drainage from washer into kitchen sink. Have drain line unclogged to take care of problem. If drain line not clogged, check for proper venting of drain lines.
A blocked pipe. The water can not go down the pipe, so it comes out in your sink.
another clog, further down the drain pipe. You need a drain snake.
You want the drain pipe for a kitchen sink at least 18" above the floor. You do not want it any closer or it might be kicked, or damaged during building or installing the faucet pieces.
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.
Take your snake on the roof and clean the drain from the vent.
A double kitchen sink that is draining slowly could be clogged with debris such as food particles. A snake is a tool that can be found at a hardware store that is used to remove things clogging flow in a drain.
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.
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.
Sounds like the main pipe is clogged. That's the big drain pipe that all three (toilet, shower, sink) are hooked too. When you plunge the tiolet it pushes back up the shower and sink instead of going down the clogged main pipe. The main pipe might need to be roto-rootered out. You could try some super heavy duty, pour in drain cleaner first. It might work.
The drains can get clogged and back up into each other. You can try plunging the drain to clear it, use drain cleaner or call a plumber.