Work, Family, Fatherland was the motto of the Etat Francais, the Petain-led regime which replaced the French IIIrd Republic in 1940, and which was itself replaced by the IVth Republic in 1944. The motto of the French 3rd and subsequent (current one is the 5th) republics is 'Liberty, Equality, Fraternity'.
The Vichy 'French State' was a different constitutional animal, and not a democracy.
The axe is a 'francisque', alledgedly used by the Franks (a Germanic barbarian tribe which would evolve into France, Charlemagne was a Frankish emperor. I know the Franks used throwing axes, but have no idea if they looked like the Vichy 'francisque'

and used as a symbol for the Vichy state.
The wheat symbolizes the fact that Vichy considered agriculture better than industry ('Earth doesn't lie'

, for reasons which were ideological (fear of the 'corruption' of city life, with attendant socialism and democracy), economical (Vichy France had practically no industry) and diplomatical (being Germany's breadbasket fitted in very well with what the Germans had in mind for France).
There were no parties. The state was called Etat Francais.
Initially, yes, but they were reminted and replaced by the new ones, which used less rare metals (nickel).
I'm not 100% sure but I think so. Theoretically, Vichy still ruled the Occupied zone, except that it was occupied by German forces so German governors could directly run some aspects of the administration (e.g. police, propaganda, etc).
Nothing. Vichy officially remained in force, except that German troops were 'protecting' it from Allied invasion, and most of the Vichy armed forces had been disbanded.
The change was 1944 when a new regime appeared (unless my memory is wrong about the date).