Great job! Especially with no prep!
- Is it possible to know how many questions I missed?
- What would my score be if I had gotten say another 5 questions correct?
It's not possible to know that. The test is not scored based upon the percentage that you get right, as our paper-based tests in school were. Most people get similar percentages correct and incorrect. The test is scored based upon a complex algorithm in which the single biggest factor is the difficulty levels of the questions you answer correctly vs. incorrectly. So it's not about getting a certain number of additional questions right in order to achieve a certain score.
Awesome on the AWA, too.
Why do you feel you need a 780+? It is EXTREMELY challenging to score 780+. Fewer than 1% of all test takers score that well. No school requires a score at that level and the GMAT score is generally considered a "gate-keeper" - that is, a lower score can keep you out, but an extremely high score will not get you in. Once you hit a certain threshold, then the schools typically just put a check mark in the "GMAT" column and move on to the rest of your application. Unless you have a very specific situation (eg, a scholarship that specifically requires a 99th percentile GMAT score - though not many do), then you may want to spend some of your time working on other aspects of your application.
That doesn't mean you shouldn't take the GMAT again, though. The one thing you may be concerned about is the 74th percentile on your verbal score. Some of the very top schools want to see at least the 80th percentile for both of the quant and verbal subscores, so you should check with the schools to which you plan to apply and see if they require that. If you do decide to take it again to raise the verbal score, you can probably get away with studying on your own (based on your starting score - though if you think you'll struggle with procrastination, etc, then you may want to set up a study group with friends or sign up for a class to keep you on schedule). You'll need to identify some resources to give you strategies for all three question types and the grammar rules for SC (this stuff will typically come from a test prep company) and some resources that give you practice questions (here, you'd want to use all of the official material that you can - the three latest Official Guide books, the GMATPrep practice tests, and GMAT Focus). Then, it's a matter of setting up a plan and sticking to it!