Learn Dutch: De hond slaapt bij de deur. - The dog sleeps by the door.

It means the dog is "by" or "near" the door. Bij indicates general closeness. If you want to be very specific and say the dog is right "next to" the door, you would use the preposition naast instead (naast de deur).

The lesson says bij can also mean "at" or "with". How do I know which one it is?

You can usually tell from the context! With an object or place like de deur, it usually translates to "by", "near", or "at". However, when you use bij with a person, it means "at the home of" or "with" that person. For example, bij de dokter means "at the doctor's" and bij mij means "with me" or "at my place".

Why does slaapt have a double 'a' when the full verb slapen only has one?

This is a great refresher on Dutch spelling rules! In the full verb slapen (sla-pen), the 'a' is at the end of a syllable, so it naturally makes a long "ah" sound with just one letter. To make the "he/she/it" form, we chop off the -en and add a -t. This traps the vowel between consonants, creating a "closed" syllable. We have to double the vowel to aa to keep that same long "ah" sound: slaapt.

Does the location (bij de deur) always go at the very end of the sentence?