When I play notes in the third register (above high C) my sound is very thin and piercing, as opposed to the warm tone we have in the clarinet's lower register. Anyone know how to fix this please?

Don't let your sound be like that! I have junior high students, and they're not allowed to sound like that... ever! We do A LOT of "register slurs". Make the bottom note louder than the second note, then don't let the 3rd register note be louder than the second. If you do that everyday correctly you'll be fixed in a couple weeks. Let me know if you're unsure what i'm talking about.

I can't comment on the firmness or shape of your embouchure because I can't see it. I can say make sure your reed is just hard enough. Soft reeds won't play up there, but that depends on which mouthpiece you use. A Vandoren "3" will work fine on a 5RV-lyre mouthpiece. A 3 won't play at all on my M13-lyre.

You can also try half-hole-ing the 1st finger for the third register.

Let me know if that helps or if you have any questions.

i think my problem is with my throat. My teacher says my throat is always "too tense", if you get what that means. Kind of like people who never had singing lessons trying to sing very high. I tried to relax my throat but it makes my embouchure relax too...

Yeah, I remember once last year I had something like that. My clarinet teacher told me that I used my neck too much and that one of my veins was popping out…and that he didn’t want to have to wipe the blood off the wall! Hahaha, so basically I just used my diaphragm more, like singers, as u said, and it went away. It also made playing easier.

Well, the diaphragm is actually an involuntary muscle, that is you can't consciously control it. When people say that, they actually mean the deep abdominals (not the 6-pack).

The air thing is right though. Make sure your air is coming from low down, and your tongue should be arched like you're saying "hee". Your air should be fast like your blowing out a candle or using a blow dart, not slow like fogging a window.

If you can, get a more experienced player to help you with your embouchure, and experiment with stronger reeds.

Make sure you're not using jaw pressure to bring out the higher register. Apply gentle pressure down the mouthpiece with the lips. At the same time push the instrument gently into your lips and jaw with the right thumb, via the thumb rest.