How can I determine if someone decided to relay through my node? Through incoming connections?

You will see that someone has offered you an HTLC and then you have offered someone else an HTLC.

Are open channels imperative to earning routing fees?

Yes, routing occurs over open channels, you cannot route without multiple open channels.

If so, how many are optimal?

It depends on how many coins you have and how man you are willing to put into each channel.