Learn Swedish: Medan huset brinner räddar vi familjen. - While the house is burning, we are saving the family.

Because of the Swedish V2 (verb-second) rule. The entire subordinate clause Medan huset brinner acts as the first building block of the sentence. To keep the main verb in the second position, räddar must come immediately after that block, pushing the subject vi to the third position.

Could I use när (when) instead of medan (while) here?

Grammatically yes, but it changes the meaning slightly. Medan strongly emphasizes that two continuous actions are happening side-by-side during the exact same time period. När just states that one event happens at the time of another, without focusing on the continuous duration.

The English translation uses 'is burning' and 'are saving'. Why are the Swedish verbs in the simple present tense?

Swedish doesn't have a separate '-ing' tense for ongoing actions. The simple present verbs brinner (burns) and räddar (saves) cover both general facts and things happening right now. The word medan already makes it perfectly clear that the actions are currently ongoing.

How do we know it means 'the house' and 'the family' and not just 'house' and 'family'?