AA JFK-LHR in A - Assume this one is the Admirals Club in JFK, but is there another we can use?
That's the only one. Fairly standard as far as Admirals Clubs go.

LHR-ATH - This is a connection from first flight. Coming in in A class, departing in D. Can we use a great arrivals lounge or are we relegated to a BA ClubEurope Lounge?
You can use the AA arrivals lounge in T3 but you must clear immigration and customs. You cannot use the BA arrivals lounge in T4. For departure, flights to Athens used to go from T4 (you should check this). You can use one of the BA Terraces lounges - which one depends on which is convenient to the gate you will depart. Ultimately you need to figure out if you have enough time between flights to use both the arrivals lounge in T3 and the BA lounge before your departure in T4.

ATH-LHR-FCO-LHR - Departure lounges in these cities, Club Europe I suppose? Do they exist in all 3 cities?
Yes
Does our Sapphire Status get us into anything else?
Not that I am aware of.

The LHR-JFK portion is all taken care of in the Concorde lounge. I cannot wait for this segment, and I am very happy we squeezed it in prior to retirement.
The lounge can get quite full if there is a full load on Concorde. If it is important for you to experience the all the facilities of the lounge (i.e. food, booze, MB spa, etc.) I would advise you to get there early.