This thing inline with the dryer vent

Before you do anything, you must figure out where it goes. It really has to vent outside. If it vents into the attic, you are going to have serious issues, including the potential of mold and rot as the hot, moist air is vented into an unconditioned space. It is (usually) against building codes to vent into the attic for this reason.

If it is venting through a roof or soffit vent, or out a wall, you're probably okay. It's best to keep runs as short as possible and with as few bends as possible. Dryer lint building up in the pipe can pose a fire hazard, so the shorter and straighter it is, the less chance there is to have anything build up, and the easier it is to clean out if needed.

Assuming it's vented properly, then I think boxing it in with drywall is a great way to go, and in fact is exactly what I did (in the corner):

You should also consider an in-line lint trap (I probably would have put one in if I knew about them):

I'd also recommend using straight vent pipe where possible, as it is smooth inside which reduces the opportunity for lint to attach and block the pipe. The rigid stuff you have is alright (way better than the white plastic crap) but it is still ridged inside.

That seems unlikely to work. A wye is not a venturi, so all that adding more air to the dryer vent is going to accomplish would be to further reduce flow from the dryer.

You are absolutely certain that the whole duct (and vent/exit) is clean and clear? You replaced the whole thing with rigid duct?