Could you substitute "vad" with "vad som" or "det som" here? It's been so long but this still feels a little vague for me. :(
Yeah I still don't quite get that either. An explanation would be very useful :)
So...."sade" and "sa" are the same thing?
Yes :). "Sade" used to be the proper word while "sa" was considered more coloquial. But "sa" is much more common now and many people find "sade" rather old-fashioned. I never write "sade" myself.
"De lyssnade inte på vad professorn sa" - can this also be written as "de lyssnade inte på det som / det professorn sa"?
Why is it på rather than till in "Studenterna lyssnade inte på vad professorn såde"?
lyssna på is the way we usually say it, lyssna till sounds a bit archaic to me.
På. It means to, in, on, and basically every other word in the English language.