The first part of your question is harder: How certain are we that the Buddha existed [at all]? The first "written in stone" evidence for the Buddha is found in inscriptions on pillars that were erected during the time of Ashoka. Dates in ancient India are not exact, but one "traditional" date for the death of the Buddha is 486 BCE; "scholarly" dating makes him slightly more recent, placing his death somewhere around 400 BCE. Ashoka would have been reigning some time around 250 BCE.

This means that our first clear evidence for the Buddha comes perhaps 150 or 200 years after his death. Aside from these inscriptions, there are of course oral traditions presenting themselves as deriving from his time. There are also archaeological sites that are associated with the life of the Buddha (his birthplace, the site of his first teaching, etc.). While these may indeed have been sites of ancient Buddhist activity, we still have not found evidence on them that pushes them all the way back to the time of the Buddha.

So, while this means that the Buddha's non-existence is theoretically possible, most historians feel that the evidence is "close enough," and scholars don't typically doubt his existence. Paul Williams has a good discussion of the issues in his Buddhist Thought (pp. 21ff). In this section he notes, "The Buddha may not have existed, although there are no serious scholars currently who take this as a significant option."

The second part of your question ("how certain are we of the supposed events that took place in his lifetime") is easier. In other words, was the Buddha the son of royalty, predicted by a soothsayer, did he see the "four sights," etc.? We have no evidence for the particular events that happened in the Buddha's life. Written sources for the Buddha's life begin around the year 0 (and of course these were based on oral traditions, but we do not have a clear idea of the history of those oral traditions). There are numerous accounts of the Buddha's life that can be found in Buddhist literature, and these are all contain different details. Thus scholars treat them not as journalistic histories, but rather stories that convey the important truths of the traditions that produced them.

It should go without saying, but just to be clear: the above is not intended to criticize Buddhism or to say that it is "fake." There is a particuar game that you play when doing history (i.e. the game of using evidence); Buddhists may have other standards, or other types of evidence, or other values, or other types of "authenticity" that are more important. Williams, around the above citation, has a nice discussion of the issues. (I.e. if the major point of Buddhist thought is to critize how "real" we think things are, how does this relate to the "reality" of his life story.)