Lions also lived in North Africa until the middle of the last century, a species called the Barbary lion. Whether or not that's the same kind that use to live in Europe I do not know, but lions in general had definitely already crossed the Sahara. Interestingly enough the Sahara itself changes on a pretty short timescale; only five to ten thousand years ago, huge portions of it were humid enough to support plant life and even early pastoral agriculture. It has apparently alternated between this state and its current dryness hundreds of times in the past few million years. We've found evidence of human habitation - bones, tools, art depicting animals and so on - in a bunch of places in the desert that just cannot support human life any more.

As for the Mediterranean, six million years ago the strait of Gibraltar closed up for about half a million years, so they could have crossed then. They also could have just swum across the Bosporus, given it's only 700m wide at the narrowest point today