Flexible duct behind dryer looks a little sketchy

If you go to your local hardware store you will be able to find a tape made for air ducts and high heat areas. It would be something like this 3M High Temperature Flue Tape.

But also a new dryer vent is probably going to be cheaper to buy than the roll of tape when you only need one piece from the entire roll. So if you can easily get to the back of your dryer (and probably you need to get there anyway to fix the hole) then go ahead and replace the entire vent. They make flexible dryer vents which easily attach to the back of your dryer and the opposite wall. (This assumes you do not have a very long run from your dryer to the wall. If you do then just go with the tape.)

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.