The university I go to..

I’d agree that “at which” is formal. I’d say the last one is very common. The first one could be used, but it sounds a bit awkward. However, if you reorder the words and get something more common, “at” is ok. “I study at Yale.” Of course, your example are not complete sentences, so this answer could change depending on the rest of the sentence.

Also, the most common would be simply: "My university..."

"makes no sense", sorry I was not fast enough to spot the auto insert predictive text feature.

If you remove the word 'where" from the last sentence it makes know sense. Unless you wish to say you are studying a particular university. This is most unlikely to be the case. that case would apply if you were to be studying a particular university as part of a very specific project or university degree etc. You are correct the sentence is correct with the word "where" included exactly as you have written.