The general difference in usage between the verbs trust and entrust is that the former usually infers a temporary or conditional handover of limited responsibility/scope that does not abdicate any inherent responsibility of the trustor, whereas the latter is a formal handover or bestowing of responsibility for something.
For example, a parent sends their children to school and trusts the school to take care of their general welfare while they are there. That trust is limited to while the children are in school, and the parent has not abdicated parental responsibility. However, you could say that the parent has entrusted the school with the child's education because the school is entirely responsible for that.
There are other examples where one is 'entrusted' with something on a temporary or limited basis but generally speaking, this describes a much more formal handover of trust or responsibility for something. For example, you have "trust" among your friends and yet there is unlikely to have been some formal handover of trust or agreement that you could point to - friends just develop trust over time, and sometimes one's trust is misplaced! But when something is entrusted, the trustee should not be in any doubt that they have received responsibility. For example, if it was said that an employee was "entrusted with managing the team" it would be formal, written into their job description. There was a specific point at which they were entrusted with that responsibility and clear definitions of what it entails.
There may also be certain subjects that could be entrusted, or given to someone in trust. Taking just one of your examples, that of a secret - you could trust many people with the same secret by telling it to them. The trust you have is that they will not tell it to anybody else. In this scenario, nobody is exclusively 'holding' that secret, so it hasn't really been 'entrusted'. However, let's say your secret is a family recipe that has been handed down through generations. You might say that you 'entrusted' that secret with one specific family member, for them to 'keep' that secret and ultimately to pass it on to a future generation of the family. In that scenario, they have been "entrusted" as it is formal, they are custodians of something, and they have a responsibility to entrust it to someone else in the future.
I'm not going to address each of your examples individually because firstly that would be akin to proofreading which is off-topic and secondly correcting mistakes doesn't teach you why something is right or wrong. Hopefully this will help you comprehend the difference between the two so that you'll be able to correct your own examples and any others you encounter.
