Re: joining aluminium gutter sections
Typically, gutter sections are overlapped with sealant between the two sections. A rivot is then usually installed at the top edges front and back to hold the joint steady. Installing fasteners below the "water line" of the gutter is ill-advised because it will become a point at which to leak.
If it were me, I'd lap the gutters into the downspout section with the sealant between the two pieces and forget about slip connectors because that just isn't aesthetic at all. Better yet would be to joint the two gutter sections and install a drop in type scupper. Drop in scuppers are either round or rectangular, simply drill/cut a hole, apply sealant around hole, drop in scupper, then use either flat headed sheetmetal screws or rivots on either side of the scupper to hold into place. In this instance the fasteners are ok because they'll be inside the downspout once it's in place.
When it comes time for the neighbor to replace their fascia, the entire gutter would be removed and set aside. Even if you put in the separate gutter section with the scupper, this would be the likely course of action by anyone coming in to do the repair. It's easier to remove an entire gutter run than to try to break a section apart and get it back together without damage to the gutters or leaks afterwards.
I suffer from CDO ... Its like OCD, but in alphabetical order, LIKE IT SHOULD BE!!!