I've referred to Kyle's post on audio demos numerous times. He's got some really slick ones!
Regarding your video editing question, it's really whatever you are most comfortable with. I personally have been using Premiere for years. Now that they've changed to a subscription model, you can just "rent" it for a month at a time whenever you need to work on your reel which I think is pretty handy.
Definitely bring your first export of your demo back into your DAW of choice to do a mastering pass after you finish the picture edits. That way you can balance all your audio, compress it a bit for an internet delivery (if that's where it's going), and also work in some nice designed audio for your transitions (or just wrangle the fades better than your vid editor can).
In terms of how to showcase video games specifically, especially something subtle like foley and ambiances, consider giving us a walk-through of your work one layer at a time. Gradually bring in other elements until you have your work within the full mix. Going back to Kyle's Saint's Row demos, my biggest critique is I would have loved to hear his materials on their own as well as in the mix (but I watched all of his demos anyway because they are hilarious and brilliantly edited). As a bonus, maybe you could split-screen parts of the demo and show off a bit of your implementation. I've demoed Camtasia (more than once!) for that purpose.