Do we have direct proof of Jesus Christ's existence from primary Roman sources? Is it possible that medieval European monks created the personification of Christ when copying Roman documents, adding him into already existing stories?

I am only going to respond relatively briefly, as my main answer to this type of question is found in the FAQ, which you’ve already been directed to. It’s here. Actually, I’m in the process of drafting an updated version of that which may appear the next time this question gets asked.

However, I do want to address this question more directly. No, there are no direct proofs of Jesus’ existence from primary Roman sources. The closest we come to that are the references in Tacitus and Pliny. There is no reason to suppose that Jesus would have made it into first century Roman sources, and that they would have been preserved.

The second part of your question is conspiratorial, because it would require an increasingly implausible historical hypothesis to sustain. If there was no historical Jesus, and he was an invention of medieval European monks, you have the very difficult job of explaining how the institution of medieval monasticism came into existence. And then you must explain how a movement like that came out of late antiquity, and how there came to be this Jesus-less non-Christianity, more or less all the way back to the 1st century. And then you must explain how those monks also simultaneously manufactured a whole history of Christianity across the Mediterranean and Europe, and how similar movements appeared in Africa and Asia in late Antiquity, and also suppressed whatever gave rise to those movements that wasn’t a version of Christianity. So you see that the whole enterprise is becoming increasingly unwieldy.

This, historically speaking, is the reason that ‘medieval monks made Jesus up’ is implausible.