Complexity of solving multiple problems of the same complexity? If I am given two languages $A$ and $B$ which are in NP, then we know that there exists a NDTM that can decide $A$ in polynomial time, and there exists another NDTM that can decide $B$ in polynomial time. Can I decide $A$ and $B$ in polynomial time using a NDTM by simply using a NDTM to check for belonging in $A$ then use another NDTM to check for belonging in $B$ and if both accept then it belongs to both $A$ and $B$?
Yes, you can test whether $A \cap B$ is in $\textsf{NP}$ by testing $A$ and $B$ separately. Actually, you have proven that the class $\textsf{NP}$ is closed under intersection. See this lecture note (Exercise 1) for details and other cases.