Looking at the Star Alliance Award Rules on UAMP, it says, with regards to stopovers:

"No stopovers are permitted on awards wholly within a geographic region. Awards between regions allow one stopover"

If I were to consider an award booking from ATH-IST-TLV-IST-ATH with a stopover in IST on the return, would that be in compliance with the above rule?

Should not be a problem... Europe to Middle East award, with a stopover... And ATH-IST-TLV is the published route, so no MPM issues either...

Is there anyway to construct a legal routing from Europe to GYD (Baku) with a stopover somewhere else in the middle east like Jordan, Qatar or Oman, or UAE? I am just trying to maximize my miles and see something else.

I can leave from anywhere in Europe as I am flying in from the USA, but I want to use the Europe-middle east award.

On what program? Every FFP has it's own rules.

I have UA miles, but thought that the *A rules for awards are the same. I also have tons of Amex points which I could transfer into a variety of carriers, but my question relates to *A awards.

On an *A award, can I construct an award from Europe to Baku via UAE/Oman or Qatar for 40K in Y?

Thanks

Unfortunately, as Kiwi Flyer says above, the rules for stop-overs are program specific, from my own experience I know that SK are very stingy, especially on awards with other carriers (with their own awards they allow a stop-over prior to/after an intercontinental sector, basically a stop-over in CPH/ARN).

As for travelling to Baku (which I was loking into this spring but eventually went for Yerevan), the * Alliance carriers flying there are LH/OS/TK/BD and none of them have any service to the middle-east which would alllow a stop-over construction. For Baku however I would also look into BT (not * Alliance but partly SK owned so you can earn SK-points), they often have very good economy fares via RIX + they give out loads of points for little money in C.