I am studying some properties related to bipartite graphs and it would be useful for me to know if there is anything known about the structure of almost all bipartite graphs. For example, is it true that

Almost all bipartite graphs of order $2n$ have bipartitons of size $n$?

Is there any known result of this type? Is there anything related to other invariants in almost all bipartite graphs (max degree, number of edges)?

Also, as far as I know there are no asymptotic enumerations of the number of bipartite graphs hence I am wondering if there are any (nontrivial) upper/lower bounds for the number of bipartite graphs of order $n,$ perhaps taking other invariants into account as well?