A conjecture of Erdős and Rényi [ErRe66], who proved that almost surely such a graph has a perfect matching (when $n$ is even).

This is true. Pósa [Po76] proved that almost surely a random graph with $\geq Cn\log n$ edges is Hamiltonian for some large constant $C$, and Komlós and Szemerédi [KoSz83] proved that \[\geq \frac{1}{2}n\log n+\frac{1}{2}n\log\log n+w(n)n\] edges suffices, for any function $w$ which $\to \infty$ as $n\to \infty$.