Learn Swedish: Han tog och skurade badkaret på morgonen. - He went ahead and scrubbed the bathtub in the morning.

This is an initiatory pseudo-coordination. Adding tog och (took and) before the main verb emphasizes the initiation of the action. It shows that he took the initiative, got around to doing a chore, or simply went ahead and did it, rather than just dryly stating the fact that he scrubbed it.

Why are both tog and skurade in the past tense?

In a pseudo-coordination, the verbs joined by och must be in the same tense. Because the scrubbing happened in the past, both the initiating verb (ta) and the main action verb (skura) have to take their past tense forms (tog and skurade).

Could I use i morse instead of på morgonen here?

Only if you mean specifically this morning (earlier today). På morgonen is used for in the morning in a more general narrative sense, pointing out which part of the day the event occurred, without necessarily linking it to the speaker's current day.

How do we know the gender of the word badkaret?

The definite suffix -et tells us it is an ett word (ett badkar). Remember that compound words always get their gender from the final word in the compound, which in this case is ett kar (a tub or vat).