Do Buddhists talk about insides? Buddhists talk about abstract and concrete things, particulars, parts and wholes, etc., and Theravada Buddhism is often I think thought of in terms of an analysis of parts, breaking things down into smaller components and the Buddhist 'dharma'.

But do they talk about "insides" at all?

I'm asking because surely everything with a beginning has an inside, but death, which must occur and so begin, can't have an inside in at least some senses, at least supposing that there is no after-life.
I think you must be thinking of 'middle'. Things have beginnings, middles, and ends. And yes the suttas are filled with discussions of all three.

"It was not until I had thoroughly understood the six senses, the origin of the six senses, the satisfaction of the six senses, the problem with the six senses, the end of the six senses, and the way to the end of the six senses, that I knew of myself that I had attained the highest attaining."

Buddhism also discusses 'inside' vs. 'outside' or 'internal' vs. 'external' which is more easily understood as 'personal' vs. 'impersonal'; subjective versus objective.