Proving properties of Random Graphs
It doesn't sound hopeful in general. 

For example, let $P$ be the statement "all vertices have degree 0 or 1". Let $p=1/n$ and $n$ even. Then conditioned on the event of being regular, with high probability the graph is 0-regular or 1-regular, so $P$ holds with high probability. Also $P$ is preserved by removing edges. But it is certainly not the case that $P$ holds with high probability under $G_{n,p}$. 

Even in other situations (say, fixed $p$ as $n\to\infty$), I would expect that there are events $P$ of the form "maximum degree less than $d$" which satisfy your 1. and 2. but don't hold w.h.p. for $G_{n,p}$. 

So something more than just the two properties you wrote down will be needed....