I have dirt or insect droppings around where the toilet meets the tile as well as piled up on one of the bolt caps. What could this be?

Sounds like you have bowl wax on the bolts and its collecting dust, and possibly around the base of the toilet.how old is your home? Might need to reset the toilet and clean area to seal with caulk.

I think truck is probably right on this one. Not to scare you, but this reminds me of a particular house I was working in. I pulled their toilet and there was a massive carpenter ant nest in the cavity of the base! That was a shocker!

Clean around the bowl and purchase a tube of Dap White Tub and Tile Caulk. Caulk around the base of the toilet bowl. (see image)
Good luck, Tom