Learn Dutch: We missen de docent. - We miss the teacher.

This is because of the consonant doubling spelling rule. The root of the verb is mis, which has a short 'i' sound. When we add the plural ending -en, it would normally break into syllables as mi-sen, turning the first part into an open syllable with a long 'i' sound. To keep the 'i' short, we have to double the 's', making it mis-sen.

Is missen used for missing a person emotionally, or missing a physical thing like a train?

It works for both! Just like in English, you can use missen when you are sad that someone isn't there (like de docent in this sentence), and you can also use it for missing a bus, a train, or a target.

I've also learned the word leraar for teacher. What is the difference between that and docent?

Both words mean 'teacher'. Leraar is very common for primary and high school teachers. Docent is a bit broader and slightly more formal; it is used for high school teachers, but also for college or university instructors. In everyday Dutch, you will hear both used interchangeably.

If I want to make docent plural, do I need to double the 't' at the end?