If English is not your first language, I apologize, but this is still a bit difficult to understand, and the specific details matter. I will try to answer what I can and feel free to add more details if I miss something.
I got a guilty notice for parking ticket and would like to appeal. guilty for lack of proof.
What proof did you give them the first time? Did you mail it, or do it in person?
1) I want to appeal my parking ticket but to a higher court and not the proceed with the nyc dept of finance appeal process. Can I appeal to a higher court?
No. NY law requires that you exhaust all your administrative remedies before you can move the case to a "real" court. If you lose the appeal though, you can file what is called an "Article 78" action in Supreme Court which would get a real judge assigned. This is way more expensive than the cost of the ticket (it's $305 just to file the papers alone) and you have to follow a lot of laws to do it right, which is why most tickets never get this far.
2) I want to file a counter claim. Is it possible to do so? which court?
Counter claim for what? Against who? Probably not going to happen.
3) Parking tickets are criminal or civil action? I want to better understand the nature of parking tickets for my research.
For parking at a hydrant? 100% civil. It doesn't go on your "record" and it is definitely NOT a "crime".
You can read more about it here (and if you scroll to the bottom of the page, it is available in other languages besides English):
http://www.nyc.gov/html/dof/html/parking/park_tickets_common.shtml
4) How do you file motions in parking ticket proceedings? example motions for discovery. motions to dismiss.
You don't. It's an administrative hearing, not a court case, so it's not allowed.
Now, here's something that might be helpful, but only if the ticket was for blocking a hydrant. Section 1202(3)(b) of the Vehicle and Traffic Law says that you cannot:
3. ...
(b) 1. No person shall stop, stand or park a vehicle within fifteen
feet of a fire hydrant except when such vehicle is attended by a
licensed operator or chauffeur who is seated in the front seat and who
can immediately move such vehicle in case of emergency, unless a
different distance is indicated by official signs, markings or parking
meters.
2. No person shall stop, stand or park a vehicle within fifteen feet
of a fire hydrant except when such vehicle is attended by a licensed
operator or chauffeur who is seated in the front seat and who can
immediately move such vehicle in case of emergency, when such vehicle
impairs or hinders the access of a fire vehicle to a fire hydrant while
such fire vehicle is engaged in an emergency operation, unless a
different distance is indicated by official signs, markings, parking
meters or the provisions of any local law, rule, or ordinance.
What that means is that it is
legal to park next to a fire hydrant
if there is a licensed driver in the front seat who can move it if needed. So, if you had a driver in the truck and still got a ticket, you can (and should) fight it based on this statute.
Good luck.