WOW! You got problems!
Some of them are no briners and I'm surprised that your plumbers or inspector didn't pick up on them.
Let me take these one at a time.
The water in the toilet in one of the not-so-often used bathroom in the basement would drain out after about a month.
This can be caused by several things.
1) A small crack in the bowl causing the water to drain back into the sewer. **OR**
2) A blocked vent pulling the water out as little every time you drained or flushed. Any gurgles heard from this toilet?
We plugged the floordrain with a red rubber plug and sealed it tightly. The smell from the drain definitely dramatically reduced
The reason it smelled was the trap seal had evaporated the seal down allowing sewer gas to escape. To climate this pour a quart or so of cooking oil or antifreeze down the floordrain.
We then entered into summer. The smell in the laundry room seem to go away for the most part
Sewer gas raises when it's warm and descends when it's colder so you would smell it more in cold weather and less in warm.
I think it comes from both the hole for the washer drain pipe and the dryer venting duct hole. I can't tell exact thought. But one strange thing is, the smell will significantly reduce if we run the dryer.
Caulk the hole with cement if through a block wall . There's less smell because the dryer is exhausting and pulling out air from the laundry room.
there is a portion of it has a black rubber connection. We guess it is for allowing for movement. However, sniffing on that rubber, we smell sewer. We are not sure if it is supposed to smell through the rubber.
Sounds like a rubber coupling was installed at the base of the stack and is now leaking sewer gas. Replace with a shielded metal band in the correct size. (see image)
we also start to smell sewer at one particular spot on the wall
Hasn't any one suggested running a smoke test on your vents and drainage to spot any sewer gas leaks? That might be a good idea.
The town utilities came out several times and been telling us there is no sewer lines going through and they have no idea why it smells.
SAY WHAT? No county sewer line? Then what are you connected to?
overall we smell it a lot more often in winter than in the summer
Same explanation, sewer gas raises in warm weather so you wouldn't smell it in the summer as much in the winter.
There is one plumbing company who suggested us to do a smoke test and would charge $1800 for it
AHH! My thoughts exactly! $1800 sounds way too high.
Get more estimates. I think you need this badly and soon. If all the estimates are that high would you like to try yourself?
I hope this answered the questions you had in mind. If I helped please rate my answer. Good luck, Tom