Possible positions for the word currently

Solution 1:

They are all correct, though I'd add a few commas here and there. The rule that dictates where adverbs should go is subtle. Adverbs go just before the word or phrase that the adverb is modifying. Essentially 1 - 3 and 5 mean the same thing. Number 5 is least natural because the modifier is at the end, but you've made it into an adverbial phrase.

At this time, we have no open positions available.

Really, there are no job openings. End of story.

Number 4 is a little different. 'Currently' is specifically modifying 'available'.

We have no open positions, that are at this time, available.

There might be job openings, but for whatever reason, they aren't being filled right now.

Solution 2:

No, there is no simple rule. McCawley discusses the facts and some theories about adverb placement in The Syntactic Phenomena of English. "Currently" is a sentence-modifier, like "probably", and your examples 1, 2, 5 are common to other sentence adverbs. Perhaps the "no open positions (are) available" in your examples is a concealed sentence, then your 3 has the adverb at the beginning of this concealed sentence (so it's like your example 1) and your 4 has the adverb after the subject of this concealed sentence (so it's like your example 2).

Solution 3:

Only number 3 reads oddly to me: we wouldn't put an adverb between the verb and the object [e.g. I wouldn't say "I threw quickly the ball"]. The others are all fine.

Solution 4:

Do you have Swan, Practical English Usage? Highly recommended, including recommendations on this very issue.

If not, peruse for now this reference at this page:

English Grammar: "(Simple, Practical yet Comprehensive) with Multiple ... By V P KANNAN