The only thing I can see in your log that I would suggest removing is
O2 - BHO: (no name) - {5C255C8A-E604-49b4-9D64-90988571CECB} - (no file)
It appears to be a deactivated browser helper object called Click-to-call which was part of Windows Live Messenger. Outside of that, I don't see any sign of a keylogger from that log. That is not to say that you don't have one, but hijackthis isn't showing one.
I wasn't but I suspect a site I was on had an ad containing a keylogger, do you see one in my log? If so it would help a lot. Plus since I don't have a credit card I'd like to not get an authenticator unless I absolutely need to.
Why do you suspect the site you were on had a keylogger?
Have you checked this page?
http://us.battle.net/en/security/theftIt may give you some ideas as to how your account may be getting hacked.
If you've gotten Blizzard to restore your account to you, I would recommend changing your password through blizzard.net. Also I see curse client running at startup, so I assume you are using a variety of addons. Malicious addons are a big reason why accounts get hacked. I would seriously take a good look at any addons you've recently downloaded and installed. If you do have a keylogger, you'll most likely find it in one of them.
And from what I have seen if you've been hacked once, then you're likely to get hacked again. So in my opinion, yes you absolutely need to get the authenticator. I know you said you don't have a credit card, but you have to be paying for the monthly service somehow. You should be able to get the authenticator by the same payment method you are using to pay your service fees.
Also, as a courtesy, if you are a member of a guild, I would talk to your guild owner or high ranking officer and let them know what's happened if you haven't already. It would also be a smart thing to remove yourself from the guild until you've resolved your account issues. One of the big reasons accounts get hacked is to rape the guild bank for gold.
To answer SpooF's question, yes normally strong password is enough to prevent your account from being hacked, but in the case of WoW there are just too many ways people can get duped into exposing themselves to malicious sites and activities. Especially if they are trying to get gold outside of normal game play. The authenticator is a security key that generates a random 6 digit number that must be entered at login in addition to your user ID and normal password. The key is associated with your account by serial number and the "random" numbers are unique to the key, hence making it virtually impossible for someone other than yourself to login to your account.
http://us.blizzard.com/store/details.xml?id=1100000822
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