The cult of Sol Invictus and Roman Mithraism were not the same thing, notwithstanding that there may have been some relationship between the two that we've yet to fathom. Mithras may have been associated with the cosmos as a whole, as opposed to Sol Invictus who was specifically a sun god. Sol (maybe Sol Invictus) appears in some Mithraic paintings alongside Mithras, so they were certainly not one and the same.
The celebration of solstices, part of the cycles of the natural world is near universal in ancient societies, and it was common to assign the winter solstice as a birth date for solar-related deities... However, AFAIK Natalis Invictus didn't itself co-opt any existing celebration in the Roman empire at that time. You need to go further back, or switch empires, to find other celebrations on this date.
It's actually very easy to determine the exact solstice date, once you've figured the position of the celestial equator (the plane bisecting the earth at right angles to it's rotational axis) relative to the constellations. The solstice is simply when the setting sun crosses the celestial equator, so you just need to remember the point on the horizon where the sun set until nightfall when the constellations are visible. Of course the Romans believed the universe revolved around the earth rather than the earth around the sun, but the position of the celestial equator is the same either way, and accurate contemporary globes mapping the constellations and celestial equator attest to the Romans' knowledge of the night sky.
I don't see any reason to doubt a historical Jesus. He was part of a tradition of wandering "miracle" working god-men, not even the most famous in his own lifetime. e.g. early Christian apologists had trouble touting their man in the face of the greater miracles attributed to Jesus' contemporary Apollianus Tyaneus, who was most certainly a historical figure... Christian apologists also cited the phoenix (bird) as evidence of the reality of reincarnation to assuage converts who had a hard time believing in the resurrection of Jesus (the phoenix was considered a real bird back then), so one doesn't need to assume a supernatural Jesus to assume an historical one.
It's very unlikely that there was any celebration of Jesus birth before he was assigned the traditional Dec 25th, since early Christians didn't celebrate birthdays - they regarded this as a pagan practice.