I actually found Henry's death indexed at the Washington digital archiveshttp://www.digitalarchives.wa.gov/
That site has indexes of deaths, but not the actual certificate.
If you don't have Henry's obit, that may be helpful. It was published in the Tacoma News tribune on January 9, 1945, page 7
Lillians was published on October 4, 1971 on page 34. There is a volunteer who will look up Tacoma obits, You can see his message on the Pierce County Washington board herehttp://boards.rootsweb.com/localities.northam.usa.states.was...
As far as using the www.familysearch.org
I find it easiest to look in the individual databases first. Scroll down the home page where it says United States, then click on that. It will take you to another page of databases.
Now as far as your Grandfather's birth, It could be a completely different Henry you found, that happened to be born on the same day. Or it could be correct, but your Henry could have been raised by an uncle and aunt, or by grandparents. It is kind of hard to say yet.
Also, do you have the marriage record for Henry and Lillian? That may list parents.
Let me know if you get Henry's obit, and I can see where you should go next.