Should it be handled differently if the OP states that the problem is fixed vs. finding an old, abandoned question that is recognizable as a hiccup?

No – if it's clear that the question is about a problem that cannot be reproduced anymore, then it doesn't matter if the OP explicitly stated that or not.

Is this an appropriate use of close question (simply marking it closed will accomplish the purpose)?

Yes. Actually, Stack Overflow has a close reason for these things:*

This question was caused by a problem that can no longer be reproduced or a simple typographical error. While similar questions may be on-topic here, this one was resolved in a manner unlikely to help future readers.

The key point is actually the last sentence (I ommitted one from the SO close reason). If the question is not useful to future readers, then it should be closed, which will eventually lead it to…

Should it be deleted on the basis that it is of no use to anyone else? (Does a question of no use to anyone else fit the definition for the site?)

… being deleted. That's what we would typically do with these questions. It's the natural way for questions to become deleted automatically if they're closed and not answered.

Now, of course, if the OP is answering their question and saying "I fixed it, there was a typo", or "I rebooted and the problem went away", then a moderator would have to delete the question, because questions with accepted or upvoted answers are not auto-deleted.

And we absolutely want to get rid of questions that are not useful for anyone except the OP. Allowing such questions would significantly deteriorate the quality of our content in the long run.

Is there some other way to indicate that the question does not need an answer? (If a question doesn't need an answer, does it fit the site's definition?)

There is no other way. Like you say, if a question doesn't need an answer then it shouldn't have been posted in the first place.

_{* Note that we cannot add another custom close reason as we're limited to three. However, seeing typos or minor "hiccups" is way more common on Stack Overflow, so it makes sense to have it as a separate close reason there. On Super User you can just flag such posts.}

Should it be handled differently if the OP states that the problem is fixed vs. finding an old, abandoned question that is recognizable as a hiccup?

No – if it's clear that the question is about a problem that cannot be reproduced anymore, then it doesn't matter if the OP explicitly stated that or not.

Is this an appropriate use of close question (simply marking it closed will accomplish the purpose)?

Yes. Actually, Stack Overflow has a close reason for these things:*

This question was caused by a problem that can no longer be reproduced or a simple typographical error. While similar questions may be on-topic here, this one was resolved in a manner unlikely to help future readers.

The key point is actually the last sentence (I ommitted one from the SO close reason). If the question is not useful to future readers, then it should be closed, which will eventually lead it to…

Should it be deleted on the basis that it is of no use to anyone else? (Does a question of no use to anyone else fit the definition for the site?)

… being deleted. That's what we would typically do with these questions. It's the natural way for questions to become deleted automatically if they're closed and not answered.

Now, of course, if the OP is answering their question and saying "I fixed it, there was a typo", or "I rebooted and the problem went away", then a moderator would have to delete the question, because questions with accepted or upvoted answers are not auto-deleted.

And we absolutely want to get rid of questions that are not useful for anyone except the OP. Allowing such questions would significantly deteriorate the quality of our content in the long run.

Is there some other way to indicate that the question does not need an answer? (If a question doesn't need an answer, does it fit the site's definition?)

There is no other way. Like you say, if a question doesn't need an answer then it shouldn't have been posted in the first place.

_{* Note that we cannot add another custom close reason as we're limited to three. However, seeing typos or minor "hiccups" is way more common on Stack Overflow, so it makes sense to have it as a separate close reason there. On Super User you can just flag such posts.}