What kind of thermometer do I use to check dryer temperature? Can same one be used to check stove temperature

a broken wire or loose connection can pass voltage but as the current draw goes up, the voltage goes down. depending on the quality of the break, the voltage at no current draw (such as with a multimeter) could vary from full voltage to almost nothing.

to really figure this out, you'll need to remove the cover of your breaker box. measure the voltage between the neutral bar on the side of the box and the incoming wires from the meter at the top. they should all read 120V or 0V. if not, then the power company needs to investigate.

also measure between the neutral bar and the two screws on the side of the breaker that feeds the dryer outlet. those should read 120V. if not, then the breaker is likely bad.

otherwise, the problem is likely a broken wire or loose connection somewhere between the breaker and the outlet.

You're mixing grease with lint, which is only going to make a mess and clog up in no time. You could also create a situation where dryer exhaust goes back into your house instead of outside, causing excessive humidity and a mold/mildew risk. The workaround for that would be to install dampers, but those would likely get jammed with the grease/lint combination. And that's assuming that this would be within code, which I suspect it isn't, or every builder would cut the corner. So, in short, my suggestion is that you don't even consider joining the two.

Also, a suggestion, only use flex pipe for the final hookup to each device, and cut away any excess flex tubing. The rest should be rigid metal venting for durability and to give you the best airflow.