In "Paul's Case," does Paul regret taking his actions in the end? How do you know?

It depends on how you're defining the end. The next to the last paragraph includes this line: "As he fell, the folly of his haste occurred to him with merciless clearness, the vastness of what he had left undone. There flashed through his brain, clearer than ever before, the blue of Adriatic water, the yellow of Algerian sands."

That's regret, pure and simple. To think of folly and things left undone is regret. However, look at the very last line of the story: "Then, because the picture-making mechanism was crushed, the disturbing visions flashed into black, and Paul dropped back into the immense design of things."

I'd say he regrets, and then, as he dies, so do his regrets.