Your question is answered near the bottom. However, I think you will find the first part interesting as it is related.
The function $\psi(x,y)$
We let $\psi(x,y)$ denote the number of positive integers $\leq x$ such that all of their prime factors are $\leq y$. Something important is that we can understand $\psi(x,y)$ very well when $y$ is fixed and $x$ grows very large, and also when $y$ grows like some fractional power of $x$.
Above you claim that $\psi(x,y)\ll \log^{\pi(y)}(x)=\exp\left(\pi(y)\log \log x\right)$. While this is true, for large $y$ it is not very good and we can actually do much better. Assume that $x=y^u$. Then we have $\psi(x,y)=x\rho(u)+O\left(\frac{x\log(u+1)}{\log y}\right)$ where the error term is uniform for $1\leq u\leq \exp\left (\log y)^{\frac{3}{5}-\epsilon}\right).$ Here $\rho(u)$ refers to the Dickman De-Bruijn rho function. For a complete survey of results regarding $\psi(x,y)$ see Hildebrand and Tenenbaum's 1993 paper "Integers Without Large Prime Factors." It is also worth noting that uniformety in a larger range is equivalent to improving the error term in the prime number theorem. This is proven in Hildebrand 1982.
The number of $n$ and $n+1$ which are both smooth: For $y$ fixed, this problem is very well known, and is exactly the statement of Stormers Theorem. This theorem says that given $y$ fixed, there are only finitely many pairs of integers $n,n+1$ where both have prime factors smaller then $y$. In fact, Stormer gave a method of finding all such pairs based on Pell equations.
I am not sure what happens if we allow $y$ to go to infinity with $x$. I tried to work it out without success, but I think it is an interesting question to ask.
Hope that helps,