Learn English – “moreover” vs “likewise” vs “in addittion” vs “furthermore” vs “as well” vs “also”, When should I use each one

In your example:

Notice that these examples fall under well-established categories of acceptable, standard English.

Think of a hierarchal chart with categories and sub-categories. In your example, "these examples" are sub-categories of "well-established categories", as far as I understand the context.

In such a chart, sub-categories are below categories, so fall under is correct.

In your second examples:

Voters fall into three main categories.
Students over 25 fall into a different category.

Here, the voters (or students) are part of (inside) the mentioned category. Therefore, fall into is correct.

There's some nuanced differences between the two. I came to know implies a more lengthy, substantial or involved process of acquiring knowledge than I learned. Synonyms of come to know include come to understand and become acquainted with, which are a little more distant from learn. For example, I came to know calculus suggests a drawn out process or a deeper, more personal relationship with the subject than I learned.

I don't know of any instances where I learned cannot be validly replaced by I came to know, but I do know that the exchange doesn't always work in the other direction. I came to know my neighbor is fine, but I learned my neighbor is unsemantic (people are not facts or ideas; you cannot learn them). I discount learned a lesson here because it carries a very specific meaning and uses learn in a different sense than usual. Cambridge even lists it as a phrasal verb (or at least thinks it merits a separate entry).

However, your question includes that after both phrases. Having that makes it about some particular pieces of information. In this case, came to know is a superset of learned, because both are completed processes of acquiring knowledge, and learning is about retaining specific facts. Because of this, I can't think of any cases where one works and the other doesn't if you include that.

Regarding your example, what's the context? Was the discovery of this information expedient and easy? Is the sentence intended to be merely informational? If so, use learned. Was it very difficult or time consuming to find out the woman's status? Is it the knowledge extremely important and worth emphasizing? Are you trying to avoid being prosaic? If so, use came to know. Compare:

I flew to Bangladesh to see my cousin. After landing, I went to a restaurant to meet her and her husband. Upon arrival, I learned that she was not in town. She had to make a business trip and would return the following week.

Tracking the billionaire's wife across Europe was no easy task, but I finally found out she was in Madrid, so I double timed it over there. Upon arrival, I came to know that she was not in town. I asked around at the likely clubs and bribed clerks at the upscale hotels, but if anyone had seen her they weren't talking.