Learn Swedish: Han stängde av telefonen.

Yes! On its own, the verb stänga means "to close". However, when you combine it with the word av (off), it creates the particle verb stänga av. This specific combination means "to turn off" or "to switch off" (used for electronics, machines, water, etc.).

Can I separate the verb and say Han stängde telefonen av like we often do in English?

No, that would sound unnatural in Swedish. Unlike English (where you can say "turned the phone off"), Swedish almost always keeps the verb and its particle right next to each other. The particle av needs to come immediately after the verb stängde and before the noun telefonen.

Which word should I emphasize when speaking: stängde or av?

You should put the heavy stress on the particle av. In Swedish, particle verbs always have the emphasis on the short particle word rather than the main verb. If you stress stängde instead, it sounds like you are just saying "closed" followed by a random preposition, which can confuse native speakers.

How do I know to use -de to make this verb past tense?