Working well

A beaver, though proverbial (as here) for its hard work, is not a farm animal in the sense that pigs, rams ('adult male sheep') and dogs are. Each of these verbs derives metaphorically from the ~ supposedly, observed ~ characteristics or behaviour of the animal, e.g. the greedy pig, the 'battering ram' (look that up!) and the dog that follows even when unwanted.
'Fly' (Answer 2) is of course also a verb and a creature, and we doubt there can be many farms anywhere without their share of flies; but fairly clearly, it wasn't the main verb here.

'To shelve' can mean: to slope gently (like a 'continental shelf, off a coastline), and also 'to put onto a shelf' (i.e., to store something away and forget about doing anything with it). 'Shelving' is a quantity of shelves (cf. 'packing / packaging', 'stuffing' etc.).

We shop at a shop (Answer 1) and mind about what's on our own, or someone else's, mind (Answer 2); bird-watchers hide in a hide (Answer 4). But we can meanwhile 'fish out' something (the idea being comparable to 'fishing', i.e. the process of getting hold of what we're after may be in some sense intrusive and/or time-consuming, such as fishing out an old photograph or financial document) ~ as in Answer 3.

'Dressing (up)', 'coating' and 'capping' are all reasonably benign activities ~ except, perhaps, in Answer 1 with the idea of hiding something or disguising it; but 'to boot' ( = to kick ) is clearly more drastic.

'Spade' isn't a verb: the word needed here is the appropriate form of 'to dig'.
One can 'hammer' information into (or perhaps, even, out of) someone, or 'hammer away' at a keyboard (punching at it more aggressively than necessary); 'nailing' is perhaps a related usage, meaning to secure or capture something so that it can't escape. ('They nailed him with a question about what he had done with the money the previous evening').
The use of tool images also includes 'screwing' money out of people (by putting pressure on them, 'twisting them' physically or situationally, or otherwise making them uncomfortable) and also 'drilling' facts into people.

Fingers, hand and head are all visible externally; but not so the tongue (Answer 2).
One can also 'thumb' things, e.g. 'a well-thumbed dictionary' (not to mention 'thumbing a lift', i.e. hitch-hiking).

'Dragging oneself' (a slangy reflexive-style verb) is clearly metaphorical; how would you, or any other ~ presumably able-bodied ~ human being 'drag yourself' anywhere, 'by the scruff of [your own] neck' or otherwise?
Meanwhile 'drag' (as adjectival noun) or the phrase 'in drag' refers ~ for reasons we probably needn't explore ~ to actors (or others) cross-dressing, usually males clothed as females.
Some people refer to a boring experience (Answer 4) as 'a drag'; this phrase used also to be used, as one among many, to mean 'a cigarette' (possibly because one 'draws', i.e. breathes in, while smoking it; or perhaps because the time taken to break off another activity and smoke a cigarette is itself a long and potentially tedious distraction from the main task at hand).

'Authoring' (as a verb) is an ugly and unnecessary usage: what's wrong with 'writing'? But at least that was a constructive stage in the process, unlike 'tailoring' (trimming, with the metaphorical scissors), 'doctoring' (raising the suggestion of painful and intrusive surgery!) and ~ most unsubtle and sinister of all ~ 'butchering'.

Water (Answer 3) is clearly [!] the only fluid, although 'buttering-up' (Answer 2) carries a strong suggestion that the butter would be melting. At any rate the phrase is heavily metaphorical ... unless there was,indeed, a very heavy and rather peculiar affair going on!

'Punch' (Answer 2) can also be a mixed party drink, rather than the action of walloping someone with a closed fist. The other aggressive actions are with the teeth ('bite': Answer 1) and feet ('stamp' and 'kick' in the lower two Answers).