Why did I get surrounded by so many treasure goblins?

This isn't common, but it can definitely happen. Every time you spawn into a floor of a Nephalam Rift, it randomly chooses which enemies will spawn on that floor. One of the groups of enemies is a lot of treasure goblins all at once. It's definitely one of the rarer enemy groups, but is fun when it happens. :)

Definitely normal. I think these probably occur in one of 20 to 30 games. They will only occur inside of a rift.

If you can kill a significant amount of them then it is usually a lot of fun (screenshot will come when I get back to the computer it is on). If you are playing above your level it can be discouraging to only kill one or two.

From what I have seen there does not seem to be a pattern to these groups forming with regards to people, levels, the type of game, etc.

edit

Screenshot as promised

And video just for fun, this was in a t13 rift