I'm not sure I can agree with any of the exceptions that you mention in your answer.
Methods not involving logic
Even if a method simply just delegates its implementation to an inner dependency, it is very relevant to verify that it happens. I presume that you write code for a reason, so you need to write a test that ensures that it stays that way.
Remember that one of the key benefits of unit tests are as regression test suites. Testing that an inner dependency is being correctly invoked is called Interaction Testing. Say that you have the following method:
public void Save(Entity entity)
{
this.repository.Save(entity);
}
It is very important to test that the repository's Save method is being invoked with the correct input. Otherwise, some other developer could come along at a later date and delete that line of code and the regression test suite wouldn't alert you.
Remember: Simple things are not guaranteed to stay simple.
Not testing database operations
Do you find it inconsequential whether data is being persisted correctly in the database? If you really, truly, don't care, then you don't need to test it - otherwise you do.
If you don't test your database operations, how do you know that your data access component works?
I am not saying that you should test your ORM library, but you should test that it is being used correctly.
Not testing object in all layers
I'm not sure what you mean by this question, but if a field can be null, and this is a problem, you need to test what happens when it is, in fact, null - everywhere.
It is much more preferable, then, to design your API so that values are guaranteed not to be null.
Conclusion
You should test everything you can test. No exceptions. Simple things don't stay simple, and you need to be able to verify that code that once worked keeps working.
There are things (such as UI) that you can't unit test, and these should be abstracted away so that they contain as little logic as possible, but everything else should be tested.
Test-Driven Development (TDD) is the best way to ensure that this happens.