Closure on regular languages
A)


  First of all because we want Reverse the accepting state would become the start and the rejecting states will become accepting, also change the direction of the edges to the oppositse side. 


You're almost there. What happens if you have multiple accepting states in the original DFA? That's where you need NFA capabilities to convert them.

Here is a spoiler in case you want to check your answer: Designing a DFA and the reverse of it on ComputerScience.SE.

B)

Your answer is true although you might want to phrase it a bit more carefully. We want to prove "if $L$ is not regular, $L^R$ is not regular". So let $L$ be not regular. If $L^R$ were regular, so were $(L^R)^R = L$, which is a contradiction. Hence, $L^R$ is not regular.