Selfridge constructed such a set with density $1/e-\epsilon$ for any $\epsilon>0$: let $p_1<\cdots<p_k$ be a sequence of large consecutive primes such that \[\sum_{i=1}^k\frac{1}{p_i}<1<\sum_{i=1}^{k+1}\frac{1}{p_i},\] and let $A$ be those integers divisible by exactly one of $p_1,\ldots,p_k$.

For the second question the set of integers with a prime factor $>N^{1/2}$ give an example of a set with size $\geq (\log 2)N$. Erdős could improve this constant slightly.