Hear here?

Answer 4 is correct: yes, you misheard 'board' as 'bored' perhaps.

'Water on the brain' ( = hydrocephalus) is of course a serious and threatening condition; but taken in a much more simplistic sense, the best way to let water run away is to open a tap (as into a bathroom basin). Meanwhile of course, 'a tap on the head' uses the other sense of 'tap' (a quick, probably fairly light blow). None of the other Answers matches 'like with like' in quite this same silly way.

The puns here depend slightly on poor pronunciation: Answer 1 offers 'Fred' as a common (and/or childish) mispronunciation of 'thread', while Answer 2 (probably slightly the stronger) has it as an elision for ' 'fraid' ( = 'I'm afraid [that] ... ' ).
Answer 3 refers to a long-running newspaper cartoon strip about a dog, but there isn't really a pun there; Answer 4 was the nickname of a banker in the news around 2010 as a ruthless manager ('shredding', i.e. getting rid of employees wherever he could).
The 'Knock, knock!' joke is a very old form of joke in English ~ usually involving some similarly awful weak pun ~ and always hinges on an unexpected name-like phrase in the final line.

'Plant' in this context means 'machinery' (as in 'plant hire' centres, where you might go to hire a concrete-mixer etc.), so Answer 3 is right. Please ignore any nightmare suggestions elsewhere!

This comes down to 'spring' (a metal mechanical component which stores or responds to tension) as distinct from 'spring' (the season when 'nature wakes up again outdoors' after the winter), though of course there are other meanings (e.g. 'to spring into action', i.e. suddenly to start moving, as though powered by a spring like the one inside the clock).
Answer 2 is therefore the best; the others are ingenious but unlikely.

The first three Doctors each have names that may seem offputting: 'Flew' sounds like 'flu' ( = influenza ), though your present writer was for several years very happily treated by a doctor with this name, long since dead himself of old age and natural causes; 'green' is the colour we associate with illness ('she was looking rather green after the rough ferry crossing') and certainly there is nothing naturally green about the healthy human body ~ except the eyes, in fairly unusual cases. 'Dr. Payne' sounds as though he may cause as much discomfort as he cures, though again there may be many perfectly good and competent doctors out there who happen to have this name and take the jokes in good spirit.
'Dr. Shipman' sounds innocent enough, except that there happened to be a serial killer of elderly patients by this name, active in Manchester several years ago. Anybody who was alive and following British news at the time would be expected to be aware of this, and no other doctors still practising under this name should fall under any suspicion whatever.

Yes, this appears to be a pun (of sorts) on 'bar' and 'baa'.
(Make what you like of this as an example of English humour ... though we bet your own language probably has similarly bizarre 'accidents' in it sometimes!)

Claudia (Answer 1) may be pronounced either as 'claw-di-yer' or 'clow-di-yer' (1st syllable rhyming with 'cow'), which gives a sound very like 'cloudier day' ~ a typical downbeat British weather reference. Carol singers (Answer 2) come round towards Christmas to sing seasonal songs in public places and maybe at people's doorsteps, possibly also collecting money for good causes ~ while indeed there may be many a holly berry (Answer 4), or more likely several, on people's cards and decorations.
Meanwhile there seems to be nothing untoward about the name in Answer 3.

The surnames in Answers 1 and 3 are entirely genuine (have a look in any phone book) but work particularly well in their contexts here: a stationery shop (not the same thing as stationAry, though presumably it doesn't move around!) sells material for people to 'read and write', and a garage will deal mainly in cars ('Kerrs'; you may not have known the slightly irregular pronunciation of this Scots family name). The bakery at Cookham (a village beside the Thames, not far upstream from Windsor, and pronounced like 'cook-'em') may well exist under just that highly appropriate name.
The odd one out this time was Lorna's Laundry at Answer 2 ~ which contains an echo, but no actual pun as such.

Yes: you saw the pictures and immediately thought 'band', and when you heard an identical-sounding word, your brain was distracted into not realising it was in fact the participal adjective from the verb 'to ban' ( = to forbid). An easy enough mistake to make in the circumstances!