I'm glad you've read up on this plan so you'll understand the fundamentals and exactly how it works.
Now, to answer your questions -
I may be wrong, but I've always counted nuts as a fat serving.
Cheese is counted as a protein, so you don't have to include it as part of your 2 cups of dairy.
And, yes, you may have unlimited servings of on-plan vegetables, along with reasonable amounts of protein. Just make sure you are eating at least 4 1/2 cups of veggies a day.
The reason it is recommended to start with 1/2 cup of beans is for obvious reasons, especially in people who aren't used to eating beans regularly, but if you can handle a larger serving than that, feel free to indulge. It is just a guideline.
On Phase 1 you should eat until you feel full and satisfied.
Good luck, and we look forward to hearing more from you!