What are the implications to consider when starting a university Flight Club?
What are the implications of starting a student Flight/Aviators Club,
  and what benefits could it provide, outside of meeting new aviation
  friends? (Implications referring to expected safety precautions a
  school might want to impose, any FAA regulations pertinent to a club,
  and associations/partnerships we could create to show legitimacy to
  the school)


I was in college not all that long ago and I can speak to this a bit as I was involved in some clubs that built cars (Formula SAE) while not the same I see a lot of parallels. Generally speaking, and this is very university dependent but I have seen it apply across the board, if a school puts money towards a club the club must abide by the school's rules but more importantly it typically falls under their insurance policy. For something like a poetry club there is not that much risk, but when we were racing cars the university would inspect them all the time and there always had to be a faculty member present when we were running tests. We were lucky in that we had good faculty backing which is key, my advice here is to first find a professor (you are generally going to need an advisor anyway) that flies and get them excited about it and onboard as this will go a long way to help your case. 

What gets tricky here is you flying on the university's dime. If you are not looking to buy a plane and you simply want to use the funds for rental that is an interesting predicament. An aviation lawyer may be able to answer better for you, but you need to confirm what the deal is with using club/org/university funding to rent and fly planes. Technically you are not getting compensated to fly but you are also not paying to fly while someone is paying for the aircraft (you are in turn paying 0 which is below your pro-rata share). If the students are splitting the rental costs, renting from a local FBO and are all checked out in the airframe you don't really need to even get the university involved.

There are universities like Embry-Riddle that are very aviation focused and offer training as a part of the curriculum so there are places that do it. 

Getting a local flight school or FBO involved may help if you have a place to rent planes from and can guarantee business they may offer a small price break.  

I would reach out to the National Intercollegiate Flying Association and see if they can offer any advice or help. 


  What would be a good way to find pilots within the University? (I'm
  sure out of 30,000 students there are some)


The airmen registry here in the US is public, you could always run it up against your school directory... I would ask around, post fliers etc. keep in mind that flying is not the cheapest hobby and college kids tend to not be all that cash liquid... I sure couldn't start flying until I had my first job. 


  Would this be percieved by the general aviation community as "not my
  place" to start a student flying club, as I am a new pilot?


IMO pilots are very receptive of other pilots. Old pilots love to inspire young pilots and you will most likely find a lot of support from them. GA awareness is never a bad thing either!