1. Crits in Beginner's are handled differently from Core, but for for now you need to first make sure the attack hits, once it hits the advantage symbols associated with the weapon is the number you need for the critical to activate. Any left over advantage can be used for other things, like passing advantage die or noting something narratively about the scene.
2. Triumph and Despair are two fold. Part A they are successes and failures that do cancel each other out. Part B, the additional awesome or awful things do NOT cancel one another out.
3. These get expanded upon in Core, but for now a Destiny Point can be used to upgrade the dice pool, either by turning an ability die into a proficiency die or by turning a difficulty die into a challenge die if targeting an NPC check. Note that the token does not "flip" until after the action.
4. EotE Core is very elegant. The narrative style of play is a different ballgame from the more simulator-based games like Pathfinder, Shadowrun, etc. People who have played some Fate or Savage Worlds are sliding right into Edge. I appreciate the d20, percentile, and other quadratic equation based games but am finding that my table, mostly 30-40 yr olds, don't have the time, will, or desire to fuss around with the math and layers of rules and just want to tell awesome stories with a streamlined dice system. It took about 10 minutes to "teach" Edge and we were playing, a few details aside of course. I'll be running Star Wars for one group and Warhammer 3E for another for a long time to come.
Good luck with your game!