Question 1) The first sentence is incorrect. The second is OK. Another way of saying this: 我是在美国出生的。The 是……的 pattern is great because it gives more structure to the sentence. Regarding the use of 里, if it was needed (which is not the case here), it would go right after 美国。
Question 2) Graduate student: 研究生, Student at the Masters level: 硕士生, Student at the Ph. D. level: 博士生, postdoctoral student: 超博士.
Question 3) 懂 is the standard, neutral word for "to understand". 明白 is often used with 了 ((我)明白了) to mean "I got it" (i.e. when you ask someone for an explanation, the person tells you, and then "you get it"). Note that when the context is clear, you don't need to use the pronoun. If you say 明白了 out loud, it clearly refers to you.