Is the white, viscous fluid that appears when rubbing the inner part of the vagina, without any external wetness, considered "madhiy" (pre-seminal fluid) that necessitates performing "istinja'" (washing the private parts) and "wudu'" (ablution) for every prayer? And is it permissible to combine prayers at the university to alleviate hardship?

You are afflicted with Waswasah (obsessive doubts), and its cure is to ignore and disregard it. You are not required to inspect what has come out of you; rather, the default assumption is that nothing has come out. If you are certain that something has come out, it could be normal vaginal discharge or Madhy (definition) (pre-ejaculatory fluid). If it is Madhy, then perform Wudu' and Istinja (definition)'. If you are in doubt, you may assign it the ruling of what you are doubtful about. You are not permitted to combine two prayers unless you are afflicted with a continuous flow of discharge or Madhy. In that case, you may combine them by praying the earlier prayer in the time of the later one, or vice-versa, according to the Hanbalis.