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Sounds like the trap in the sink is getting the water pulled out of it. It probably wasn't vented properly.
In the bathroom like every other kitchen sink
Maybe sewer gas backing into your drain. Call a real plumber. This can actually make you ill.
Upgrade your cycle. Use both cups in dispenser and fill with detergent. Use fresh detergent. Dishwashing detergent powder does not have a long shelf live. Pour detergent in a zip lock bag to store.
You apparently have a creative drain system in your house. Your kitchen sink and your bath tub drains are tied together on the same branch flowing to your main line. I am assuming the toilets are not bubbling or overflowing when the kitchen sink drains. If they are, your main sewer line is clogged. If not, you need to have someone (preferably a plumber) run a cable or snake, as some people call it, through the kitchen sink drain.
The trap in the drain line under the sink is dry. It should have water in it to stop the smell
That would be a p-trap, like what is located under every sink. This U joint holds water to prevent a sewer smell or odor from the sewer or tank from coming back into the house. It is a trap of water that prevents air gas from reeking your house out!
Check to see if all the pipes are on and make sure there is not a open clean out drain there if there is make sure it is close
If indeed it is IN your sink, simply run lots of hot soapy water through it.
It would be my suggestion that you need to have the sewer "snaked" out as there is an apparent obstruction in the run from the bath area to your septic tank or sewer line and the the kitchen is "up-stream" from that bathroom, or, perhaps the shower is on the back side of your kitchen waste and the obstruction is associated with the showers drain and waste run...i.e., a hair clog, etc, is building up and combining with the waste material from the kitchen. Perhaps there is a clean-out plug under the kitchen sink which you can run a small "snake" through.
When the kitchen sink backs up in the tub, this usually means that your sewer line is stopped, clogged or blocked up between your house and the asasive tank or your asasive tank is full.
How big is the kitchen sink