Name the screenwriter

No idea

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Looks like Gary Busey in the top shot and Michael Angarano in the middle shot...that's all I've got.

Correct on both.

And Rufus Sewell and Kiefer Sutherland in Dark City.
Hider in the House + Haywire + Dark City =
Lem Dobbs






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Pretty sure the bottom shot is from The Duchess and the Dirtwater Fox, but the screenplay for that film is credited to three different people.

Definitely The Duchess & the Dirtwater Fox. The top is Crimes of Passion. The middle one is The Other Side of Midnight, which doesn't help all that much except the IMDb notes this fella did an uncredited pass. So if you count that it is...
Barry Sandler






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He didn't actually write any of the Another 48 Hrs. screenplay. He just gets a "characters created by" for having co-written the original, but...
Bad Dreams + Another 48 Hrs. + Commando =
Steven E. de Souza






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No idea

I'm pretty sure that these are all Disney movies, but I don't know who the common link is.

I think the first picture is Bob Crane, but I don't recognize the movie.

The second picture is Jim Dale in the movie Hot Lead and Cold Feet.

The third picture is Don Knotts, and maybe Kim Richards, but I don't recognize the movie.

The last picture is Kurt Russell, maybe in the movie The Barefoot Executive.

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Must be Joseph L. McEveety?

You are correct. Claire

Pauline Collins is all I got

And that's Hal Linden holding the newspaper in the top shot

It is Pauline Collins and Hal Linden. Here's another film by this writer.

Hesper Anderson?