Can 'sendChat()' accept the same string as clicking on a character sheet?

I am trying to script the equivalent of clicking on a character sheet. Is there some special formatting or options that I need to pass?  For example, I send "/me does a thing to @{selected|token_id}" and what I see is the emote formatting of "PintAndPie does a thing to @{selected|token_id}".

I'm not sure what you're asking here.  @{selected|...} and @{target|...} will not work when issued by the API.  You'd need to pull the selected tokens out of message.selected array, and there is no @{target|...} equivalent.

Basically, I want to cut&paste what I see in the UI chat history and re-run it through the sendChat() script method. Not possible?

It's possible, but you almost certainly need to make some edits to the text for it to fit the script context.  As Aaron observes, any selected or target keywords will need to be changed for the relevant id or character name (which the script can do), and there might be other changes needed.

If you're talking about manually copy/pasting the command, you'll get it with all the attribute references and such which can be converted to make a generic version the API can issue.  If you're talking about the API observing messages in chat via on('chat:message,...), it will only see the result after substitution (no @{} references, etc). If you look at my OnMyTurn script, it has a function to deal with fixing up character ability/macro text which would probably be what you need, or at least a starting point.