no, amiga 1200 has version 3.0 kick start roms and AGA chipset, but most old games are written for 1.3 rom and OCS/ECS chipset. You need to use a utility like relokick to load 1.3 kick rom and then possibly you can play those games.
about your mouse, probably either the mouse port or the mouse have some hardware problem. You better try it in workbench environment though.
