How to use the validation rules on both client-side and server-side?
No, no such thing exists.
Yes, you need to code it yourself.
However, I imagine you could possibly create widgets which are able to deliver generic validation js routines based on the modelFields.  Such as "This should be chars not more than max_length".  However they could not trivially generate client-side code to validate any custom validation written in python, nor even something like "This must be the username of an existing user". But if you take this trouble to build widgets which do basic validation based on the modelFields please contribute them to open source :-D
I'm using jQuery validation system for client-side validation. The backend works with django.
jQuery use an interesting set of rules in JSON format.  
Does exists something to use the same rules on django side or I need to code it myself?