Honestly, don't even think about SQ F on the A380 because it's not even available on paid *A RTW trips.

You might get lucky with space opening on SQ C A380 from JFK-FRA but it will tough to get. I recommend getting EF (ExpertFlyer) and putting in an alert since it's definitely worth the money in doing so.

What I recommend is this:
1. EWR/JFK-ZRH-BKK-DPS on LX C/F - LX C/F - TG C
2. EWR/JFK-FRA-BKK/SIN-DPS on LH F - LH F/TG F - SQ C*
3. JFK-DOH-BKK-DPS on QR C - QR C - TG C (I'm assuming you're using US miles which means this isn't possible)

*You won't get access to the FCT in FRA if you take the TG F segment from FRA to BKK because it isn't on LH, LX, or OS.

I also recommend flying out of JFK instead of EWR if you're flying LX of LH F due to the better lounges especially the LH one.