Moreover, can you extend a ground wire?
Technically, you should twist the extension bare ground wire to the short one and secure it with one of the copper crimps like you have pictured. A wire nut certainly would work , but it should be green. Practically speaking, either way will work and be safe, just a matter if you want to follow current NEC standards.
Similarly, do I need to ground a subpanel?
The neutral and ground MUST NOT be bonded at a sub-panel. They should only be bonded at the main service panel. The ground wire from the house must be connected to the sub-panel ground. Since it is a detached building the ground rods are also required by code and should be tied to the sub-panel ground.
To tap into an existing wire with a connector, simply cut the cable where it needs to be spliced. Strip both ends as well as the end of the new cable you want to add. Place a wire nut of the appropriate size onto the three cables. Twist the wire nut in place and ensure that the wires are not coming out.