I used any type of contact adhesive but mostly Evo or any of trade equivalents Dunlop Thixofix was good not seen it for years, I always double glued my raw chip edge, do it once wait till it drys and do it again then I do the piece to be stuck to it. Any mistake can be easily corrected by purchase of Evo cleaner that removes the glued trim when applied on its glued edge. I have had to do whole sheets when alignment on some Gondola Triangles misaligned on a rebate. This was a repair and usually I would have glued the face sheet and then rebated, in this case it was bloody hard getting it to land on pre trimmed edge even with winding sticks as the edge I aligned against was proud of surface.
Edge preparation you can trim to a closer size by trim cut on a table saw (if you find this difficult on your saw put a sacrificial piece on the bed to allow the trim piece to slide over easily) or formica hand cutter (but you need to snap the cut by hand but I would not do it myself as the file does not need it that close trimmed but I have got out of jail in many occasions cutting bits from sheets by this method) but I mostly used a combination of trimmer cutter on one of them small trim router (in my shop) or a file (when on site and and finishing mostly the end cuts on worktops, they usually supply edge materials in the length of worktop packaging, I just glue/stick trim with file). Practice is required with the file as you will find it can take off the trim very quickly or it snaps off to solid edge like butter, it is dependant on quality of laminate and you have to take care when down on the face edge and allow for a small back bevel to prevent cuts or snags from finished trim. But this is the same for all crafts there is always a bit of skill to be learnt.
The file is a smoother and ordinary metal type, not a rasp or one with large filing teeth. Would not give you the finish you desire. Watch the different action when you lead with the filed edge or smooth. But a wee practice would not go amiss to get the technique right before doing a finished job. Good luck.