Mick,
Firstly, welcome to HT. I'm sure you wll find this an invaluable source of information and I hope you find the members here as friendly and helpful as I have.
Secondly, don't be surprised if you get a visit from the big bad mod (they get really upset about using upper case all the time). No doubt you'll get away with a friendly word from Briar.
As to your question. Yes, flights do run from Hurghada to Cairo (not sure about Hurghada to Luxur though). Regardless, be ready for some early mornings or late nights as the internal flights seem to be timed for the most inconvenience.
Anyway, enjoy HT (I know I do).
WG