Am I correct in thinking that it would cost more if you used your own agent with Sonnet 4 than going through Claude Code since you would have to go through the Anthropic API? What models do folks with custom agents usually use? And what kind of prompts seem to provide the same responses that Claude Code would give you?

Claude Code is free if you have a subscription to Claude. It's much more affordable than using the API key directly.

Doubt it, based on my ccusage output. I do about the monthly charge in tokens in a day.