Am I supposed to give the Lexicon back to Septimus once it is transcribed? I did and he said he would send for me when it was translated or whatever, but I've been waiting for weeks and have heard nothing—or should I have taken it straight back to Paarthurnax?

You take it to Septimus. If you are under level 15 he'll say hell send word for you. Once your level 15 a courier will drop off a letter from him. If you are 15 and over you can just take it to him and he'll give you another quest.

Q: What do I do with the transcribed Lexicon?
A: Eat it.