A: No. The 2960 (non-S) uses different images, typically c2960-lanbasek9-tar.xxx . Using a 2960S image on a 2960 will brick the device.
The file is a Cisco IOS Software image designed specifically for the Catalyst 2960-S series. c2960s-universalk9-tar.152-2.e9.tar
Switch# show license
: Legacy. Stable. Dangerous. Useful.
: The "universalk9" designation indicates a cryptographic image that supports both LAN Base and LAN Lite feature sets, allowing for versatile deployment in enterprise or branch office networks. c2960s-universalk9-tar.152-2.e9.tar