Why do I so often want to replace "happen" in sentences?

Solution 1:

Occurs invokes the concept of a definite start or beginning. It sounds better to you because it usually gets at a more precise meaning of the concept that you want to communicate (based on your examples).

"The sound always occurs at midnight."

Happens is very similar but does not invoke the concept of the events start or beginning.

"Stuff happens."

It is subtle, but happens would be more vague in the way that you used it.

Here is a quote that supports this:

Occur is more formal, and while it is generally fully interchangeable with happen, it may be more specific in implying the time or action of an event.

Solution 2:

It strikes me to be a formal/informal distinction – I would not describe a scientific phenomenon as happening, but as occurring.