Project Directors are selected based on their ability to hold or create the vision for a project, motivate and inspire the team, and their ability to focus the game to the vision and the game pillars. Any individual on a team who has demonstrated these qualities at a senior level as they've risen through the ranks in their discipline (usually to lead status) is considered a candidate for Project Director. Rich Taylor, for example, consistently demonstrated strengths as a lead programmer, and also demonstrated good judgment and decisions on how to go about making the game he was leading (Mask of the Betrayer, Storm of Zehir, and now Dungeon Siege).
Lead Designers are selected for much the same reason - they're usually senior designers who've shown the same strengths in upholding the game vision, ability to motivate and lead a team, and can manage effectively. Like other lead roles, Lead Designers are not necessarily chosen for their design ability, and they may not be the best designer in their discipline, they simply need to understand the design pipelines, understand the toolset and its breakdowns, and how to manage a team - this is because leads spend more time managing the designers in their discipline than doing actual core design work. If they excel in design, as Josh does, then that's a bonus.