I've been listening to the soundtrack album on Spotify. Here's how I understand it:

"Her whiskey is good and her figure is neat"

Americans might not be aware that in Britain, "figger" is the standard pronunciation of "figure." Americans regard "FIG-your" or "FIG-yer" as standard and "figger" as dialectical, hence the spelling.

"The parts of every gentleman do stand up at attention."

I think it's unlikely to be "pants" because Brits do not use "pants" to mean "trousers" as Americans do. In Britain, "pants" means "underpants." Therefore it wouldn't be noticeable that someone's "pants" were standing at attention unless they were sitting around with no trousers on, which is unlikely.

"Pretty Kate of The Garter and Star down in town"

Probably just a mishearing or mistyping by the original poster.

"Fat Dolly who keeps The Red Hart of renown"

The White Hart (i.e. stag, or male deer) and The Red Hart are common emblematic names for British pubs. "Renown" should not be capitalized because it is not part of the name of the pub.

"As does the part that lies between her left toe and her right toe"

"Path" is possible, but I prefer "part" for no particular reason.