1. Corn is a grain not a vegetable, so keep that in mind. Generally a number of primal people will eat it when it's in season because they enjoy it, but if you're just starting out, you might want to cut it out completely and then add it back in later so you can truly assess how it affects you. As for fruit
2. See above on corn, except switch to white if you're going to have it. Also, read the book, or any book on primal as soon as you can! It's not just the gluten that's the problem with grains, it's the undigestable parts that destroy your intestines (read "Wheat Belly" if you really want to be convinced).
3. I buy once a week but I don't eat a huge variety. I can find some frozen so I use those as much as possible, though my primal options are pretty limited (asparagus, peppers & broccoli are the only primal veggies I've found frozen).
4. Only you can know that. Buy as much as you want, and make the space for it. If you still don't have space, cook some immediately and start saving up for that mini-fridge. If you ARE going to get a mini-fridge, make sure you read a lot of reviews for the model you're interested in. The ones I used while in university did not keep anything cool for long. Milk went bad within 3 days so I can imagine that it wouldn't be good for veggies either.
5. You can make it work. I don't have a lot of money to spare after rent, student loans, and utilities, but between me and my boyfriend, we can make it by on around or less than 75 USD/week. Depending on the price of groceries in your area and how much you eat at home (ie. someone else buys), you could probably make it work. When you can't afford top quality meat & veggies, the regular stuff (canned, frozen, not grass-fed etc.) is fine and still a far cry better than pasta.
Welcome aboard and happy recent-birthday!