(Q1) It depends whether the two predictors are highly related (in which case they'd be strongly correlated). If they are, they'll overlap in the way they help predict the outcome. In such a case, p-values are not the soundest way to choose betwen these predictors. One needs to make a selection based on extra-statistical criteria if one cares about explaining the outcome.
But if they're not highly related, and one has an odds ratio far from 1 and a small p-value while the other's odds ratio is close to 1 with a large p, then most good researchers would consider the latter variable to be noise and would consider a solution that preserved it (or many predictors like it) to be overfitted. It would not figure to crossvalidate well; it would be capitalizing on chance and would ultimately perform more poorly than a "leaner" one.