Even though only Daven named the theme, I think everyone kinda had it.
1. Two co-stars: One won an Academy Award for playing a mentally disabled person, one won an Academy Award for his directorial debut.
All the President’s Men
Solved by Melville (comment #1).
2. To find room in the budget for a social service program, the President persuades the Secretary of Commerce to stop a series of Public Service Announcements about cars.
Dave
Solved by Tom Hilton (comment #2).
3. “Nice thing about you, Joe, is that you can sound like a liberal, but at heart you’re an American.”
The Best Man
Solved by George (comment #6).
4. Dressed as playing cards at a costume party.
The Manchurian Candidate (1962)
Solved by Melville (comment #1).
5. Groucho Marx has an uncredited walk-on cameo—his last screen appearance.
The Candidate
Solved by George (comment #6).
6. “My guys are not stupid. They always put the big Jews on my schedule.”
Bulworth
Solved by George (comment #6).
7. The candidate tells a completely fabricated story about his WWII veteran uncle, and cries every time.
Primary Colors
Solved by Hogan (comment #8).