I do not start editing a question just to remove a "Hello" in the beginning and a "Best regards" in the end. But when I'm already editing a question for other reasons (formatting, spelling, grammar) I often remove those unnecessary lines. 

They do not help the quality of the question and they're also not really improving the tone of a question as you said in my opinion. It's nothing serious, but simply unneeded. And from my experience, the shorter and less noisy a question is, the better are its chances to find an answer. This is simply because most people out there are lazy and don't want to read too much. The difference is not huge when you just remove a greeting, but it's still unnecessary lines that make the question longer than it would need to be.

Also, your line from the example:  
> PLEASE HELP I CANT USE MY COMPUTER. PLEASE HELP ME.

This is horrible and should not be tolerated. 

1. It's CAPS LOCKED which means the OP is shouting at us.
2. It's simply repeatedly *begging* for help.
3. It does not add any value to the question but is the most emphasized sentence in the entire post.

For these three reasons, I would delete the entire line if it appears like that in a real post. And while I'm at it, the greeting and thanks would fall as well.

By the way I just collected a set of tips for users that tend to get negative feedback on their questions. I discuss all the issues mentioned here and many more over there:  
https://meta.askubuntu.com/questions/15152/tips-for-users-to-write-better-questions-and-avoid-negative-feedback