...stepping into someone else's shoes, especially Web Design and / or WordPress development, is never a recipe for success.
That said, and fully understanding that you may have no choice in the matter, and that you've taken on the responsibility, here are some thoughts.
Q: "Since I have DLed via FTP every file on the host associated with this domain, does that count as a database backup?"/B]
A: No.
"I am walking into an abandoned site design by another designer, hosted by an ISP"
There is a major difference between a "Website" and a WordPress blog tinkered with by your predecessor.
If you do a little Googling you will see a number of excellent tutorials on what makes up a WordPress site..
You have a base installation, including the database (this is what needs to be updated), plus any plug-ins, hacks, design tweaks, etc, plus any content your client added.
Hosted by an ISP? Why? There are Web Hosts who charge $5 a month, and for that you can have WordPress as a "one click install" that you can set to automatically update itself every time there is a new release. Unless you enjoy doing this work, you'd be much better off moving to a host that offers that.
"The owner of the site (not the designer that bailed) told me she has not done the WordPress upgrade because she is afraid she could lose her files"
What files? If it's the base installation, so what? Install it again! If it's a custom skin, it's in the "Themes" folder...save that folder and install it again.
"My question on the WordPress forum had sunk with no comments or views within an hour."
That's probably because the people in that forum aren't your personal help desk, because WordPress is a skill that takes time to master, and that anyone who has that skill and is qualified to answer your question is expecting you to go out and do your homework, just like they did, rather than answer you "within an hour."
Trust me...people in this business sometimes have a thin skin and the easiest way to guarantee no one will get back to you is to post a message saying "I posted this and no one got back to me."
Best of luck to you.