My best guesses are as follows:
1) You are correct; Sulin is a town in the Presov region of northern Slovakia.
2) "Aparious" is probably a corruption of "Eperies", the German name for the town of Eperjes, now Presov, the administrative seat of Presov region.
3) "Flareous" may be a further corruption of "Fragopolis", the Latin version of the same town (Presov).
4) On the other hand, I checked JewishGen's All Hungary Database, and it's possible the town name may be Szulin, which is in Szepes county in Hungary, and for which there are a number of Drechsler/Drexler entries in JewishGen.
5) In your search, remember to consider Hungarian/Yiddish versions of their Americanized names, though there are no rules and their original name could have been anything. But in general, Anna may be Netti; Edward may be Ede or Eduard, Henry may have been Chaim, or Henrik, or Heinrich, or Hanoch, Mor may have been Moses or Moric or Moritz, Cecilia may have been Tzilli or Cilli or Czaeczilia, etc.
6) It sounds like you are working mostly off of oral histories, which is great, but you need to verify that information with documentation--death certificates, naturalizations, census records, etc. There are a number of websites with searchable databases. If you can tell us where you have already searched, we can make more focused suggestions for you.
7) In your search, try "sideways" research. If you can't find any documentation for Edward, try to find as much documentation as you can for his siblings. Their records may give you the information you need.
Best of luck,