Help – keep garage dry and well-ventilated (dryer exhaust hose)

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.

I'm presuming you're using the pillow case as a lint filter. The bigger problem will be humidity. You'll be putting all of the moisture from the wet clothes into the air. After a couple loads, this makes the room really, really, REALLY humid (I've been there).

If you can swing it, you might be well off getting some flex dryer duct and running the end out the window while you're doing laundry. Then you won't need the pillowcase at all, which might cause some other problems due to back pressure/poor airflow (e.g., it might take longer for your clothes to dry, and it might cause the dryer to run hotter).