No, the device does not have a recording function such as USB recording or timeshift. Several sources, including test reports and product specifications, explicitly confirm that these functions are missing.
Yes, the Sharp TV 55GP6260E can be mounted on the wall. It is compatible with VESA wall mounts, but you need to ensure the mount is suitable for the TV's VESA pattern and weight. The specific VESA mounting dimensions and weight of the TV should be checked in the user manual or the manufacturer's specifications to choose the correct wall mount.