2 small Japanese pull saws, a western push saw, fret saw, set of chisels, bit brace with a roll of arbor bits, rabbet plane, side rabbet (trim) plane, 2 shoulder planes, 3 set of diamond stones, slip stone, multiple files, a rasp or two, 2 small bar clamps, a mini vise, hand scrapers, scratch beader with cutters, combination plane with cutters, smoothing plane, jack plane, block plane, combination square, steel straight rule, 12′ tape measure, small bottle of glue, a few short dowels, 2 marking gauges, and a few different marking instruments (awl, pencils, marking knife).
Two things which I took to instantly and now wouldn’t want to do without are an old-fashioned two foot, four fold ruler which gets used all the time (there’s nothing that comes close to it) and a 1 1/2″ butt chisel which gets used all the time for marking anything which needs to be chopped e.g. the side of a mortice. It also means fewer chops and therefore a bit more accuracy when cutting out a knife wall.
It is my understanding that frame saws are standard to a continental toolkit. Richard’s list of a hard-point handsaw, 10-12” backsaw, and coping saw is a very standard British/American toolkit, that preforms the same roles as the frame saws you detailed. All three are available in every hardware store in America and I assume in Britain, true hardware store back saws are now junk and to get something new that preforms as well an old Disston, like Richard stated, you have to upgrade to likes of Veritas and Lie-Nielsen.
Wooden mallets are mostly used for hitting your chisels when cutting joints (like dovetail joints or chopping mortises). You should never hit a chisel with a metal hammer. Build or buy a mallet that is made of fairly hard wood (e.g. maple, oak, beech wood, etc.), that will feel well balanced in your hand, and that gives you enough weight to chop with your chisels.