Learn Hardcore Croatian: Rekli su mi da će povrat ići na karticu čim budem poslala paket natrag. - They told me that the refund will go to the card as soon as I have sent the package back.

Rekli su mi means They told me (3rd person plural). Croatian often uses a vague they (e.g., customer support, staff) without specifying who exactly.
If a single known person told you, you’d use singular:

Rekao mi je = He told me
Rekla mi je = She told me

What case is mi in Rekli su mi and why?

Mi is dative (indirect object), meaning to me. With reći (to tell), Croatian typically marks the person receiving the information in the dative:

Reći nekome = to tell someone → mi/ti/mu/joj/nam/vam/im

Why do we use da after Rekli su mi?

Da introduces a content clause (a subordinate clause expressing what was said), like English that:

Rekli su mi da… = They told me that…
In Croatian, da is very common and usually not omitted the way English often omits that.

What is the tense/structure će ići in da će povrat ići?

Će + infinitive is the standard future tense (Futur I):

ići = to go
će ići = will go / will be going
So da će povrat ići literally means that the refund will go… (i.e., will be sent/credited).

Why does Croatian say povrat će ići (the refund will go) instead of something like they will refund?

Croatian often uses an impersonal/process-style expression for payments and transfers:

Povrat će ići na karticu = The refund will go to the card (i.e., be credited)
A more agent-focused version is also possible, e.g. Vratit će novac na karticu (They will return the money to the card), but the given phrasing is very common in customer-service contexts.

What exactly does povrat mean here?

Povrat literally means return, and in this context it means refund (return of money). It’s used a lot for online shopping returns:

povrat novca = refund
povrat alone is often enough when the context is clear.

Why is it na karticu and not na kartici?

Because na changes meaning with case:

na + accusative = movement/transfer onto/to → na karticu (to the card, i.e., credited to it)
na + locative = location on/at → na kartici (on the card, location)

So money being credited is treated as “movement” → accusative.

What does čim mean, and how is it different from kad?

Čim means as soon as / the moment that and implies immediacy.
Kad means when and is more neutral.
So čim budem poslala paket natrag = as soon as I send the package back.

Why is it budem poslala? What tense/mood is budem?

Budem + past participle is Futur II (often called the “future perfect” in explanations), used for an action that will be completed before another future action.

budem poslala = I have sent / I will have sent (from the future point of view)
Here it fits perfectly: the refund happens after the return is sent.

Why is the participle poslala (feminine)? What if the speaker is male?

In Croatian, the past participle agrees with the subject’s gender and number.

If a woman says it: (ja) budem poslala
If a man says it: (ja) budem poslao
Plural: budemo poslali/poslale, etc.

What is the role of paket and why is it paket (not paketa etc.)?

Paket is the direct object of poslati (to send). For a definite single object in the affirmative, Croatian uses the accusative:

poslati paket = send a package
Paket happens to have the same form in nominative and accusative (inanimate masculine), so it looks unchanged.

Why is natrag at the end, and could it be placed elsewhere?

Natrag means back and is an adverb. It often comes at the end, but Croatian word order is flexible:

…poslala paket natrag (very natural)
…natrag poslala paket (possible, more emphasis on back)
The meaning stays the same; placement mainly affects emphasis and style.