This is such an important and profound question. I am not sure if it was answered here before, but I will try my best to answer, and may Allah guide us all.
First of all, no, having doubts does not make you kafir (non believer). 1
Usually university life poses a lot of challenge for the thoughts and different arguments about religion and what not. I was doubting things about Islam before, but there is something that I always believed in; no matter how I doubt this and that, there is something I couldn't doubt; that there is a God: Allah.
No matter what this and that says, this life can't be without Allah. It is just that simple. So what the other says is either wrong, or that we don't understand the subject matter well, thus the arguments are flawed (while we might not realize how they are flawed.)
So take it simple. We have taken Shahadah (the testimony of faith), and thus we believe in Allah and His prophet Muhammad (PBUH). And from there, we believe in His angles, books, messengers, the day of judgement, and predestiny.
If you have doubts in any of these, then you should renew your belief, by reading more about each and every of the above mentioned pillars of belief.
In Islam, there is nothing that contradicts reason. Moreover, Allah calls people throughout Quran to his way by invoking them to contemplate, think, and reason. 2 3 However, there are things that are unknown to us, and we should believe them as revealed and submit to them. We don't know why we were born to our parents, and not other parents, and we don't know why we have to fast in Ramadan, and not in another month, and so on.
In our lives, we might know something is true, but not sure how it is true, and that does not mean it is not true. For example, we might know that a medicine X is good for this and that by trial and observation, but not sure about the exact workings of it, and that does not mean it is not healing. In Islam, we might not know really certain things for sure, either because we don't have enough knowledge, or because it is made unknown to us (Al-Ghaib), but that does not mean we are going to deny Islam altogether.
In addition, before we question Islam about something, we need to question, if the other side is actually correct and valid. We should realize that we are the products of our environments, and there are things we are taking for granted, while they may not really be correct and valid. Just because everyone is doing this or thinking that, it does not mean it is correct and valid.
However, there is one point here to mention. We should realize and differentiate Fiqh issues apart, as some of these issues are to do with a specific context of a society or a nation, especially in matters that are not specified by a clear text.
Moreover, realize that science changes, and what might be true today, might not be tomorrow under some conditions that we weren't aware of. In addition, look in your society, and you will see some irrational things that are accepted and taken for granted, while with some thinking, you might arrive to realize how irrational they are.
When I am in doubt I just remember this verse:
O people, an example is presented, so listen to it. Indeed, those you invoke besides Allah will never create [as much as] a fly, even if they gathered together for that purpose. And if the fly should steal away from them a [tiny] thing, they could not recover it from him. Weak are the pursuer and pursued. 4
Go on on your studies and pursuing, and inshallah, you will find barakah and success in it. Never fear your doubts, and remember, there is always Allah no matter what everyone says or does.