Is "divisible by 15" the same as "divisible by 5 and divisible by 3"?

Yes, if a number $n$ is divisible by $15$, this means $n=15k$ for some integer $k$. So $n=5(3k)=3(5k)$, so it is also divisible by $3$ and $5$.

Conversely, if $n$ is divisible by $3$ and $5$, it is a simple lemma that it is divisible by the least common multiple of $3$ and $5$. Since $3$ and $5$ are coprime, their lcm is just $15$.

Yes. $15|n \implies 3*5|n \implies 3|n \text{ and } 5|n$. Conversely, $3|n \text{ and } 5|n \implies 15|n$. This is because if $x|n$ and $y|n$, then $\text{lcm}(x,y)|n$ where $\text{lcm}(xy)$ is the smallest number that is divisible by both $x$ and $y$. In this case, $x=3$ and $y=5$, so $\text{lcm}(3,5) = 15$, therefore $15|n$.

(Note: $a|b$, read as "$a$ divides $b$", means that $b$ is divisible by $a$.)