A short at the output of a signal generator, for sure will not damage a generator. By construction, any generator should be able to generate full signal while shorted indefinetly without any damage. In RF this is a frequent situation. The so called 50 ohm impedance output is in fact a series 50 ohms resistor.
Not sure what you have in mind when saying reversed power. About pushing power into the output of a signal generator, it depends how much power. If not too much, it might survive.
For example, my Rigol DG4202 is auto-disconecting its output when voltage from outside is detected.