What's better: clemency or strictness?

As I have mentioned in other questions I am teaching a neurobiology lab course. And it is horrible.

A tiny fraction of my time/energy goes into actually teaching, and the rest is taken up by dealing with problems generated by students failing to do what is asked of them:

1. I had to fail a student for plagiarizing.

2. On the first day a student asked to be excused from the quiz because she had been on a humanitarian trip the previous week.

3. One student turned in a late assignment for no reason.

4. One student turned in a late assignment because she claims her computer spontaneously rebooted overnight and she didn’t save it.

5. One student turned in a late assignment because she injured her foot in a car accident five days ago (and gave me no advance warning that she might have trouble completing the assignment).

6. One student argued with me for ten minutes over an assignment worth 0.5% of his grade (half a point) that he swears he turned in. I have no evidence of this and cannot find it anywhere.

So far I have taken a hard line with all of them, but they are starting to wear me down. My logic is that it will only help them in the long run if they are held to the standards set out for them. Am I being unreasonable? Can anyone make a case for clemency?

All I want to do is teach the kiddos about brains. But they are making it really goddamn hard.