A game that you'll enjoy

The paraphrase and word order in Answer 3 are acceptable, but flow marginally less clearly (it is better to put the prepositional relative qualifying phrase after the 'many' or 'several' ~ or 'some', or whatever the other quantifier happens to be).
Answer 2 is wrong because 'these' (while indeed referring back) is not a true Relative word, and the comma on the join between two main clauses does not offer substantial enough punctuation (though as with many more such subtle points, you may freely come across poor examples 'out there' from native writers who ideally should have known better).
Answer 4 makes sense, but the clash of register between 'of whom' (formally correct) and 'loads' (very informal) undermines the supposedly solemn overall disciplinary effect of the passage.
You would need to avoid slipping into 'many / several / most / some of WHOM ... ', which would tend to suggest that the clients were themselves in some way the fault of the staff!

'That', 'which' or no link-word at all are each acceptable in English.
Speakers of many more precise languages may find Answer 4 surprising, but it's all right; while our NOT accepting 'the car who ... ' may seem reciprocally strange!

None of the words offered after 'in' in Answers 1-3 works correctly here, though they might well each be understood. 'In which' would have been fine; but why not, just, 'where'?

We seem to be narrating the potential turning-point of one person's life ('hers'), if not of two of them; so even if we drop out any relative words in their actual spoken dialogue, we perhaps feel the need to tell the outline of the story in relatively formal, solemn (and certainly correct) English. Answer 3 is therefore our best option, although if you were telling this story aloud to others ~ for some reason ~ you could just as well find yourself adopting versions 1 or 2.

You cannot, in good English, have an Answer-1-style blank as the Subject of a relative clause, as here ~ though it is fine to represent the Object.
'Who' and 'what', for different reasons, are both unacceptable here, though you may well come across 'what' doing such a job in informal and/or dialect situations (e.g. 'working-class' mechanics discussing the engine of a tractor).

You may quite often hear examples like Answer 4 in everyday English speech, but the Direct Object form of 'who' is, technically, nevertheless, 'whom' as in Answer 3. Meanwhile, as above, Answers 1 & 2 (as they appear here) are at least equally acceptable options.

Don't forget the difference between 'hearing from' people that you were previously in touch with, and 'hearing of' someone like an artist or composer, whose work you may never yet have recognised. ('Have you heard of Charles Dickens?')
The other Answers on offer here were each misconstructed in some way or other.

Answer 3 is also possible, particularly in speech; but Answer 4 is more clearly correct.

'What', here, is short for 'what thing' or 'what cause'; but English never needs to say such things in full.

Answer 4 is correct but pedantic, and would be considered pompous and eccentric in most normal social circumstances.