Are the cover's sides supposed to be glued or solvent welded?

No.  The cover is built using inTank Tab Technology™.  The cover is cut from a single piece then precision heat bent.

What if I need a notch or custom cutout for my equipment?

Easy - The cover is easy to cut with a pair of hand snips, rotary tool, razor blade or anything similar.  The ABS plastic is easy to cut and will not shatter - it is very soft.  You can cut an area to custom fit your particular equipment.