Learn Danish: Jeg planter et træ i haven i dag. - I am planting a tree in the garden today.

You could, but Danish generally relies on the simple present tense (planter) for both 'I plant' and 'I am planting.' At this level, you know the er ved at construction, but you would only use jeg er ved at plante here if you wanted to heavily emphasize that you are right in the middle of digging the dirt at this very second.

The sentence ends with i haven i dag. Can I swap these and say i dag i haven?

No, Danish has a strict word order rule for descriptive phrases at the end of a sentence: Place almost always comes before Time. Because of this, the location (i haven) must sit before the time (i dag).

How does the sentence change if I want to emphasize that this is happening today by starting with i dag?

If you move i dag to the beginning of the sentence, you trigger the V2 (verb-second) rule. The subject and verb must swap places so the verb remains the second core element in the sentence. It becomes: I dag planter jeg et træ i haven.

Why do we use the preposition i for the garden (i haven) instead of på?

Danish typically uses i (in) for locations that are considered enclosed spaces or clearly defined areas, such as a garden, a forest, or a city. You would use på for open surfaces, islands, or specific public institutions.