Mission accomplished! At least partly so.
We finished at 1:30 AM, after about 9-10 hours of work, give or take some chatting, some smoking of some kind of horribly smelling carnation cigarettes (not by me), some cooking (by me) and some eating.
I have mostly good news and some questions.

We followed the two videos I mentioned above. The amount of dust and dirt! My goodness! Here's a small example:

Everything was cleaned with rubbing alcohol, reading glasses cleaner and a special vacuum cleaner attachment my friend brought. We replaced the keyboard. We scraped off and replaced the thermal compound. $11 for a tiny one-inch munchkin of a thermal compound tube at Radio Shack? One inch, if that!
Reassembling was perhaps the most challenging part of the job. They are no detailed videos or directions I could find, so I would rewind the disassembly video by a few seconds, play it and try to do that segment backwards and so forth. A pretty crazy-making endeavor, I must say. If you ever want to punish someone -- I highly recommend this method.
My friend used the wrong screws somewhere inside and so at the very end we were having trouble with a couple that were too short. Also,we were having trouble putting in place the right side of the speaker grill, it was too high and wouldn't let the monitor come down. My friend didn't want to reopen everything, so eventually he managed to coax it into place. There is a small problem though, not sure if it's related or not. The right side of the monitor sits slightly higher on the hinge than the left one, making it a few millimeters askew. No one will probably notice it, but I can see it. The left side of the monitor is much too tight, in my opinion, because it doesn't pivot easily. Any suggestions on how I can fix it? BY MYSELF?
Next problem, maybe related to the short screws, used on the outside or maybe not (pure speculation on my part) is that the touchpad is a lot less responsive. I played poker on Facebook, to repeat the conditions that made the laptop get too hot before, and I lost a few hands,

not through bad play, mind you but because I clicked on a move that I wanted to make and it did nothing. Sometimes it works fine, sometimes I need now to hit the touchpad a few times to make it do something. Same question as before.

Now the very good news:
No more ringing at all, the computer boots to Windows!!!, the cursor is no longer frozen when I start, the new keyboard works great! It boots very slowly, but it does get there. Maybe now you can advise me how to speed it up, if it's even necessary. I counted about 7 seconds between the HP logo to Microsoft progress bar, which runs for 15-20 seconds, then it goes dark, then completely black for a second and takes 5 seconds to get to Windows screen, and then it takes some time to get to the desktop screen. Something along those lines.It would be nice to figure out what makes it lag so much on a reformatted hard drive but this can keep. BUT... all the big problems - gone. I am still missing the Bluetooth driver and Java plugin crashes a lot, as well as Shockwave Flash, but this was never an issue that was part of my laptop "going crazy".
The temperatures:
Last night, as I mentioned, I played a game on FB, after we were done and my friend was gone - the laptop got quite hot again. Not as hot as before, but the Speedfan showed most values with a fire icon next to them. It was 49, 48,54,52,51. Still, much lower than the high 70s before. I did have it sitting on a Belkin fan pad (I think, it was 2:30 AM) and definitely had Speedfan on automatic control.
Today, I searched online and found someone suggest that it helps a lot with the heat problem if you change the Maximum Processor state to 99%. I did that. The minimum was set to 5% and I didn't touch it.
The temperatures today -
a lot better than last night, no fire signs. I don't know why it is, but I'll be very excited if it stays that way.

I have many more questions to ask but this War and Peace was long enough, so I'll keep it for the next time, after I hear back. Thank you, everyone, for your help and encouragement!
Edited by Sophia L, 04 June 2012 - 09:38 PM.