Learn Russian: Тань, скажи Кате, чтобы она закрыла дверь.

Тань is the colloquial truncated vocative form of Таня. In spoken Russian, when you address someone using a first name or kinship term ending in -а or -я, you often drop that final vowel. It makes the speech sound faster and more casual.

Could we use the truncated form Кать instead of Кате in this sentence?

No, because the truncated vocative is only used when directly addressing someone, like calling out to Tanya. Katya is not being addressed directly here; she is the indirect object of the verb скажи (tell). Therefore, Катя must take the dative case ending, becoming Кате.

Why is the verb закрыла in the past tense if the command means "to close" the door?

In Russian, when you report a command or express a wish using the conjunction чтобы, the following verb must be in the past tense. Even though the action hasn't happened yet, the structure чтобы + past tense is the required way to translate "tell someone to do something."

Why doesn't the direct object дверь change its ending here?

Дверь is indeed in the accusative case, but because it is a feminine noun ending in a soft sign (-ь), its accusative form is identical to its nominative form. Only feminine nouns ending in -а or -я change their endings to -у or -ю in the accusative.