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1a THE French FBI agent is one doing the donkey work (3-3). Is it leg man and why?

12a Stick on gallows, up is a difficult situation. (3,4) Gum tree fits but why?

le ( the in french) gman (fbi agent)

FBI agents are known as G-men hence Le(G-man)

stick = gum
tree can be synonymous with gallows, for instance the 'Tyburn-Tree' was the name of the gallows built near Marble Arch.
and up a gum tree = in a difficult situation

Thank you so much Sudokulover and Trevor. I've now finished the crossword, although it's taken since Saturday.