Do it Yourself - Reflexives

If you chose Answer 1 we suspect you may be French ...
Answer 3 is the best version. English people don't need to say that they 'wash themselves', and they tend to do it 'in' (rather than 'under') the shower. These other expressions may well have a logic of their own, but they simply aren't what English people say. We tend to 'have a wash' or 'have a bath' instead of 'showering / bathing ourselves'.

Lots of other languages use reflexive verbs to say what people call themselves. (Do YOU ever 'call yourself' by that name, when you're talking-to-yourself inside your head? We doubt that, somehow ...)
English just says 'Someone's name is X' (or, less usually, 'X is my name').
If you know the famous musical (and film) 'The Sound of Music' - starring Julie Andrews, whose diction is such a wonderful model if you are learning English - you may remember the song 'Doe, a deer ...' in which the third line goes 'Me, a name I call myself'. Well, that's true, and it does work in English; but have you ever heard any English speaker actually say it?
(We know, 'Je m'appelle Marie', 'Mi chiamono Mimi' etc ... but that isn't the English way!)

The cats (and colonies) are plural, so we need to start from 'them'; and the plural form of 'self' is 'selves'.

Simple enough, and not a 'self' in sight! (Even in the bathroom mirror!)

Again, there's no need for any '-selves' here. If, in English, someone 'hides' (for instance), we assume they are hiding themselves - unless we are told clearly that they are hiding something else (like a secret, or a key or money that they're not supposed to have). In English we can 'wash and dress' (etc.) rather more quickly than many other languages, without any loss of understanding!

English people certainly don't need to 'walk themselves'. Answer 4 might be possible, in that we 'go and do something' ('Go and have a look at the weather'; 'She went and bought a new dress'); but 'going AND walking' doesn't quite seem right.

We don't need to 'relax ourselves'!

We only 'look at ourselves' in the mirror (usually), and who would want to spend a year doing that? So the verb-phrase we need here, instead, is 'look after' ... (meaning, 'take good care of'). We know that the speaker is talking to more than one person because he (or she) says 'everybody', so we need the plural reflexive form 'yourselves', rather than 'yourself' (individually).

'By itself' (or, in other circumstances, 'myself / yourself' etc.) means 'all alone':
'I went for a walk all by myself.' ( ... by the polluted stream, or elsewhere!)
In this case we need 'itself', because the machine is neither masculine ('himself') nor feminine ('herself') as it might be in some other languages.

In these three examples, the 'reflexive' is needed every time, though some English speakers might not say 'himself' in the last part of the sentence.
But we hope you will now recognise that in far more cases than not, English does NOT bother to form a 'reflexive' in the same way as many other languages. The reflexive element is unsaid, but automatically understood: if I say 'I wash', without mentioning what I am washing (such as my clothes, or car) ... what else CAN I be washing, except myself?
In some ways, English is simpler ~ and still, clearer ~ than you may think!