'Put yourself in her side' I have never heard before, and we don't use the preposition 'in' with the word 'side' in this way. 'Put yourself on her side' would mean support someone in an endeavor, especially an adversarial one, in which whoever 'she' is might need support. Usually, though, we would use the expressions 'take her side' or 'get behind her'. 'Put yourself in her mind' could be taken to mean 'understand how she thinks about things'. To be 'on someone's mind' means that the person is thinking about you. We would not say 'Put yourself in her perspective' but rather 'see things from her perspective'. Hope this helps.

No. None of the variants say the same thing. Being at someone's side and being in someone's shoes are just two different things. Look at it from her perspective. Look at it from her side of things. Look at it from her way of thinking. All three mean "Put yourself in her shoes."

in her side (no) on her side (yes) in her mind (yes) on her mind (no) in her perspective (no) The best way would be to say "from her perspective".