Edit: I see from the comments that $b$ in part 2 is restricted so that it can't be the least element of $B$. That's fairly important information, and changes my answer.
One hint will apply to both parts equally well. The most familiar form of a Boolean Algebra is the power set of a set. Consider the power set of an infinite set to answer both of your questions.
Added: Your example--letting $B$ be the power set of the natural numbers--works as an example for both. As you pointed out, its atoms are precisely the singleton subsets of the natural numbers. Hence, $B$ has an atom--in fact, infinitely-many, but it has at least one, which is what matters--and so the answer to Question 1 is "yes." On the other hand, given any non-least element $b$ of $B$ (that is, any non-empty subset of the natural numbers), there is at least one atom less than or equal to it--for example, the singleton containing only the least element of $b$. Hence, the answer to Question 2 is also "yes."