Moderator Action: Moved to General Discussions
Land units will heal automatically by some small amount (10 HP if outside your culture borders) if they engage in no action on a given turn. They will not heal if you are constantly moving them and/or using them in other ways.
The setup screen should be showing you all available leaders (the drop-down menu shows a handful of leaders, but can be scrolled down to display others). If that is not the case, I would recommend validating your game files through Steam.