Walls – How to cover seams in plywood walls

In general, the shorter the vent, the better, so moving the dryer next to the basement wall would be the optimal solution. If you can't do that, then moving the vent as you said would be a lot better than your current set-up.

This article covers the whole topic in a lot of detail, see especially the sections on why to keep the length as short as possible:

• Dryer Vent Hose Length Considerations
• Dryer Vent Hose Lengths: Keep it Short

Running the vent to the new location shouldn't be a problem; my preferred tool for cutting the hole would be a hole saw, but they can be expensive. Looking at your photographs, it seems like don't have any plumbing or electrical stuff to complicate the job.

Start by clearing out any loose particles that are still clinging to the wall.

Once you've got it cleaned up, wood filler will take care of the missing plywood patch. Apply it with a putty knife and sand it smooth after it dries. It will be easiest if you use a putty/taping knife wider than the patch you're repairing. Otherwise, you'll have to do a little more sanding to smooth down the ridges created between passes - either way, it's not a big deal. (If you've already got a smaller putty knife, I wouldn't go out and buy a big one.)

After sanding, wipe the area down to remove all dust, then apply a coat of primer.

After the primer dries, paint the patched area. Try to feather out the new paint overlapping the original. It looks like you've got plain white semi-gloss or high-gloss paint here. Don't forget to match sheen as well as color, unless you're lucky enough to have some of the original paint left over. One or two coats should be fine, especially if you primed beforehand. Also try to use the same method in which the paint was originally applied - it looks like a brush was used originally, so go with a brush for this. A roller will leave an 'orange peel' effect, which will stand out from the surrounding area.