What are the differences between the verbs
Could someone explain the difference(s) to me, please.
Passieren is informal (it's also what children would be using), vorkommen still a bit informal. Geschehen and vorfallen feels neutral to me, sich ereignen and stattfinden more formal. Sich ereignen might sound old-fashioned.
Vorfallen is almost always used for serious negative events, passieren for less serious but still negative events, and vorkommen will sometimes refer to negative events as well. Sich ereignen and geschehen feel neutral, stattfinden a bit positive.
Stattfinden is for something which is planned and which will take some time, e.g. describing some festival. Passieren and geschehen imply a rather short timeframe, almost a point in time, and a random element.
Vorkommen describes something rather rare, passieren perhaps a bit more frequent but still exceptional. Geschehen and sich ereignen do not imply any frequence one way or another. Stattfinden suggests that it might be something recurring. Vorfallen is probably referring to something that is occurring more often than it should, no matter how often that is.
Note that this answer concentrates on the almost common meaning between these words. Some of them have different meanings as well, for which none of the others would be synonymous, like “eine Grenze passieren”, “das Öl-Vorkommen begutachten”, “das Geschehen aus der Luft betrachten”, …