1: Mettre la puce à l'oreille
éveiller les soupçons.

2: dissimuler, mentir, ruser
Three near-synonyms appear close together: dissimuler, mentir, and, a few lines down, ruser. It is important to make sure that you use three different English equivalents, and it is acceptable to use either single words or short phrases, e.g.: sneak around, lie, cover up, and so on.

3:Avoir la langue trop longue
similar to 'avoir la langue bien pendue', i.e. 'être bavard'.

4: Eprouver quelqu'un
le mettre à l'épreuve, le soumettre à la tentation.

5: j'ouvre...
Note that in this anecdote, the speaker slips into the present tense, as often happens in ordinary speech. It is appropriate to reproduce the tense-changes in your translation (even though the published translation does not do so) as long as it is done in a way that corresponds to normal English usage.

6: Ce qui se sera passé
ce qui a dû se passer. This is a standard use of the future perfect, similar to the English 'I know exactly what she will have done.'

7: Assister à
être présent à, participer

8: Désoeuvrée
inoccupée, sans rien à faire

9: Nom
here must refer to the Christian name (normally 'prénom'), given the context.