I think, from the content of the aya it is clear that suspicion is to be shunned when it may maen harm and injustice to somebody else.
The nuts with your father I would not call that suspicion. Either he throws up when eating those nuts or not. If not you do not have to suspect anything unless there is a medical condition for it, and if has thrown up at another times. Just weight the likelihood bearing all facts in mind. You are not suspecting that your father does it on purpose, do you? Because THAT would be wrong, but not wrong that youhave an aproximate idea of what could happen, that is caution.
The same with the key to your treasure. If you feel at easy putting in a key, do it. You are not pinting neither litterally nor figuratively a finger to anybody. It is just a general measure of caution. Someone may open the safe just out of curiosity, inadvertence, and then you might get really suspicious. If you feel that somebody might directly be suspected without reason, then it is for you to weight, the hurt to you may inflict that person and your own peace of mind.
I think that the aya points to not saying a word to anybody if a suspicion crosses your mind. Keept it to yourself. I think the qur'an is social mores is mostly common sense. We know what gossip is, we know when a person is confiding us something because he or she need to confide or vent his or her stresses or anguishes, and when he or she is merely giving in to the need of "eating somebody else's flesh". We should listen and try to apease those who simply need to get out of themselves what anguishes them without harm nfor anybody else and to stop and discourage those who want to gossip never mind whether someones reputation or peace of mind is at stake. Eating somebody else's flesh is a very eloquent description of the deed. It is a matter of compassiona and justice. We are all poor wretches many times, we should differentiate between a need of somebody who feels unsafe or misunderstood or not understood and the desire to point spears at somebody else while feeling safe and spitting venom.
We should be cautious with everything we do and say. Not suspicious. Cautious, mindful of the harm something may mean to different people.
Salaam