/u/genosgrande - the G7 is, by far, the best video camera on your list - and a pretty good still camera as well: https://www.flickr.com/groups/2808623@N20/pool
As far as lenses go, I recommend a $499.95 used G7 body with the 14-42 kit lens and a 3 year store warranty from Henry's to get started - plus, when budget permits, a $307.89 Kipon Baveyes Ultra AF or $649 Metabones electronic focal reducer/adapter and Canon glass.
That way, you won't have to do an expensive lens mount "changeout" if you decide to upgrade to a cinema camera or interchangeable lens camcorder at some point in the future.
Hope this is helpful and good luck with your decision!
/u/MrGonghen - if I was starting out, I would get adapted lenses. Here's why:
Pros: access to thousands of Canon EF mount lenses - you won't have to buy new lenses if you buy or rent a Canon EF mount camera (e.g., Canon Cinema EOS, Blackmagic URSA, Panasonic LT, RED Raven), equivalent lenses cost less money (e.g., this $110 50mm f1.8) - autofocus is available with an autofocusing adapter - focal length reduction and an extra stop of light available from any Canon EF lens with a Metabones Speedbooster or Aputure DEC LensRegain.
Cons: Autofocus is slower with adapted lenses, Canon APS-C lenses are larger and heavier than equivalent micro 4/3 lenses
With a $600 budget, I would buy a $245 Kipon Autofocusing adapter and something like this $350 used Sigma 10-20mm f3.5 for Canon mount.
Here is an example shot with the G7 and adapted lenses, to include the Sigma 10-20mm f3.5:
Hope this is helpful, good luck with your lens upgrade and best of the holidays to you and yours!