Hi Louise! Triathlon training is much more fun than just running
Have you picked your first event yet?
In answer to your question, yeah, I train several mornings a week starting at 5am, it's a fantastic way to start the day, especially at this time of year.
It's the only way to fit in enough sessions, work full time and have two small kids, oh and lunchtime training helps too.
I tend to try to roughly alternate days between cycling and running, but don't follow it exactly.
If I'm doing more than one session in a day, I don't tend to repeat the same sport in the same day, for example, if I cycle first thing, then I would run or swim later in the day, but wouldn't generally cycle again that same day.
Hope this helps