Q: WILL APPLE DO ANYTHING ABOUT MY POWERBOOK 5300CS WITH BROKEN HINGE AND RELATED PROBLEMS?

No. It's obsolete. Apple will not service them any more.

I don't know the exact time frame but I'm pretty sure Apple does not inventory parts for computers obsoleted 20 years ago like yours. Their current practice is to drop support for products according to the terms set forth in this support page:

Vintage and obsolete products - Apple Support

I have dozens of old non-working Macs lying about and it would be nice if someone would wave a magic wand and make them eligible for free warranty service again. I believe in the Tooth Fairy but not in Apple supporting my ancient Macs.

Those Macs are over 20 years old, you will need to find a third party to repair them. Apple does not support products forever.

http://www.welovemacs.com/paupmapo53.html

http://www.applemacparts.co.uk/store/advanced_search_result.php?keywords=powerbo ok%205300&search_in_description=1&sort=1…

Or there may be a computer museum that can help you find parts or someone who can help with this old hardware.