First... "nas" sounds more natural here. Nas = in, on, at the; às = in, on, at, to the... Basically... we need much room to discuss about it, but it is a question of prepositions that will become clearer over time...
This hopefully is a simpler question (though I look forward to learning the answers to the longer one) - is there a difference in pronunciation of á and à?
Actually, some people use to pronounce "à" like "double a". However, the accent shows it was linked, so there is no need to pronounce it like two a's.
Há = a = à = á (same sound)
The point is that when you have "á", it is the stressed syllable of a sentece:
BÁR - ba - ra
ÁR - vo -re
Is "nas segundas feiras" acceptable? It is certainly used in continental Portugal.
Why can "Mondays" be written as "as segundas-feiras" and in other places, for example, as "nas Segundas." Why do you drop the "feiras"?