Can Javascript write to local hard drive (besides cookies)?
Not really. However, as HTML5 support becomes more widespread you'll be able to use things like Web Storage and Web SQL. You won't be able to write arbitrary files on the user's hard drive, but using those two technologies you'll be able to persistently store and access data.
Can Javascript web application run
with Javascript capabilities in
browsers turned off? (I would think
outright no, but an article on
Adaptive Path said 'maybe')
It really depends on how you define "web application." You can write web apps without using Javascript for anything but UI candy, in which case you can degrade gracefully without without it. However, it's also possible to write web apps that rely heavily (entirely, even) on Javascript, which will utterly fail without it.
Is AJAX illegal to use due to Eolas
patent claim? Is it worth it spending
effort learning it when the future is
not secure?
I'm not a lawyer, but I'd agree with the other answers -- you shouldn't worry about it. I'm certainly still writing AJAX stuff :)
Is Javascript mainly concerned about
user interface i.e. smoothen
interaction between application and
users and not about logic processing,
number crunching or form processing
etc.?
It's about both. And more than that.
Javascript has really come into its own in the past few years. Browsers have gotten a lot faster at executing it quickly, and people have been figuring out new ways to use the language itself to its full potential. You can really start using Javascript like a full-out application programming language, and not just to write little scripts that animate something or validate input.
If you're just getting back into the language and haven't read Crockford yet, I would highly recommend it. It's a great starting point to realizing the full potential of Javascript.
Edit: Some good Crockford Links
Javascript: The World's Most Misunderstood Programming Language
Javascript: The Good Parts (This is a presentation. Crockford also wrote a book by the same name that I haven't read myself, but I hear it's quite excellent.)