Is the usage of the word "fine" correct in the following sentence?
This works fine
It seems that the author of the statement wanted to say that something was working, but not working well.
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I believe that in case of written English, the sentence does not indicate the mood of the author. It's more related to spoken emotion here. The author may grudgingly say 'This works fine' to indicate it's not as well as he had expected, or in an exciting tone, he may say 'This works fine!' to indicate his pleasure.
Yes, that usage works fine.
It is most likely to mean that it is working well enough, conveying satisfaction/approval (though without enthusiasm). You might say something like this when asked about replacing something you are satisfied with. For example:
Developer: The front-end is done, but it's kind of ugly. Need me to fix it?
Telemarketer: Get our new super-fast DSL! Only $299.99 per second!