I'm going to answer your questions in reverse order.
Is it a completely idiomatic structure, or is there some way to connect it to more regular grammar?
While these individual examples are idioms, the basic principle is completely regular. Compare:
- We work nights.
- We work long hours.
- We work weekends.
- It's raining bucketloads.
- It's raining ash.
- It's raining men.
The last one is the only one where we can't just swap in another noun phrase without changing the syntax: this is a hint that "going at it hammer and tongs" is short for something like "going at it with hammer and tongs". In other words, it's shortened from a prepositional phrase used adverbially.
How do you analyse them?
In the first case, a NP is used adverbially to give a duration of time: same as "I slept all day" or "I was in Canada last week".
In the second case, an NP is the direct object of the verb "rain".
In the third case, a PP is being used adverbially to give the manner of the action.
How do these phrases composed of nouns end up being used as adverbs?
Different ways for different phrases, in the end!