The tenses in all your examples are perfectly fine.

I believe that in the last example you could have switched to present perfect if you wished so it would look something like:

Mike asked why you didn't come to the party with me, but I explained to him that you have been working hard for the past two weeks, and now you're free all you want to do is get some rest.

(I hope I got my present perfect right, I'm not too hot at grammar terms.)

Note that you probably want to say 'but I told him it was because...' or 'but I told him that it was because...'. In the final example you wouldn't normally have 'me' twice in the first part, you'd say 'Mike asked why' or 'Mike asked me why you didn't come to the party, but...'. In casual speech you'd probably get away with it, though.

It's perfectly legitimate to switch tenses as you speak, even within one sentence, PROVIDED that the time frame has changed.

To take an obvious example: "Yesterday I wore a blue shirt but today I am wearing a red shirt." I switch the tense from past to present because the first half of the sentence is talking about yesterday, the past, while the second half is talking about today, the present.

Of course there are more subtle and complex examples. Consider, "I didn't eat lunch yesterday because I have been on a diet." I begin with a simple past because I am describing one discrete event: lunch yesterday. But then I shift to present continuous because the diet is a continuous thing. I didn't just go on the diet yesterday, skip lunch, and then decide to quit dieting. (Well, that's how I usually treat diets in real life, but this is just an example.)

Most of your examples fall into that category: you're shifting tenses because you are, indeed, talking about different time periods. "I told him that I work for HP." "Told" is in the past tense because you are referring to one discrete event in the past. But "work" is in the present because, at least at the time you told "him" this, you were working for HP in the present. The sentence, "I told him that I worked for HP" is potentially ambiguous. Is "worked" in the past tense because you worked for HP at the time you made this statement, but you no longer work there now, when you are telling this story? Or was working for HP in the past at the time you made the statement? Either interpretation is possible and it would take more context to make clear which is meant.

Note that you should definitely NOT switch tenses where there is no change in time frame. Every now and then I read something where the writer carelessly changes tenses for no apparent reason, like, "Yesterday I went to visit my friend Bob. He isn't home. I called him on the phone. He answers and says ..." etc. There's no reason to shift tenses there because all the events are happening in the same time period. At the least it's disconcerting, at the worst it could create confusion over when events happened.

Also, is it possible to use both past and present forms, when only one part of a sentence is still true?

Mr. Brown asked why you were absent, but I told him that was because you are ill.

It would be "more grammatical" to say ...but I told him you were absent because you were ill whether the person is still sick in bed or had recovered, but especially if the person were no longer ill. It's not that the original phrase isn't carrying the tune, to speak figuratively, but it's (grammatically) pitchy, dog.

Mike asked me why you didn't come to the party with me, but I explained to him that you had been working hard for the past two weeks, and once you're free, all you want to do now is get some rest.

Either:

... and now that you're free, all you want to do is to get some rest

or

...and once you were free, all you wanted to do was to get some rest.

Once you are free = when you become free. So you can't say "now".

Once you were free = when you became free.