do we at least have $Y_n=\Theta(\frac{\log n}{n})$ almost surely?
The answer to this is no. It is not even true that $Y_n=\Theta(\frac{h(n)}{n})$ almost surely (a.s.) for any function $h$ such that $h(n)\to\infty$ (as $n\to\infty$). Indeed, suppose the contrary: that $\liminf_n nY_n=\infty$ a.s. Then, by the Fatou lemma, $$\infty=E\infty\le\liminf_n EnY_n=\liminf_n n\frac1{n+1}=1,$$ a contradiction.