theres a mix knob on the omega that will give you mix between playback and input so you can directly monitor whatever you are playing along with what ardour is playing.
Yes, get a decent pair of headphones and use the 1/4 inch jack on the interface. Lexicon Omega works flawlessly on my Debian laptop (Lenovo ThinkPad), 4 in - 2 out. Route your Ardour master channel output to Omega and that’s about it.
Also, make sure the USB connector you are using for the Omega works properly and your Omega is not behind a USB-hub, sharing a connection with a flash drive or something.
You don’t mention how you are managing your connections with Jack server - from Ardour or Qjackcontrol? Cadence?