Jag tror att hon inte hittar bankkortet hemma, men hon letar ändå.

Both are possible. Att is the subordinating conjunction meaning that, and it’s very common after verbs like tro (to believe/think).

• Jag tror att hon inte hittar... = more explicit, very standard in writing and careful speech.
• Jag tror hon inte hittar... = also correct and common in speech (and fairly common in writing too), just slightly more informal/less “marked” as a subordinate clause.

Because after a subordinator like att, Swedish uses subordinate clause word order: the negation inte normally comes before the finite verb.

• Main clause: Hon hittar inte bankkortet. (verb before inte)
• Subordinate clause: ... att hon inte hittar bankkortet. (inte before verb)

Not in standard Swedish. Hittar inte in that order is main-clause style and sounds wrong inside an att-clause. The natural version is:

• Jag tror att hon inte hittar bankkortet hemma.

They’re different actions:

• hittar = finds / manages to find (successful result)
• letar = looks for / searches (the process, success unknown)

So the sentence contrasts “She doesn’t find it” with “She’s still searching.”

Swedish often uses the definite form when referring to a specific, known item in the context—here, the (particular) bank card she’s missing.

• ett bankkort = a bank card (indefinite)
• bankkortet = the bank card (definite)

Yes, it’s one word. Swedish compounds are usually written together:

• bankkort = bank + card
  Then the definite ending attaches: bankkort + et → bankkortet.

• hemma = at home (location/state)
• hem = (to) home (direction/movement)

Here it’s about location: she’s not finding it at home, so hemma is correct.

Ändå means something like anyway / nevertheless / still. It signals contrast: despite the expectation that she won’t find it, she continues the action.
So men hon letar ändå ≈ but she keeps looking anyway.

Yes, ändå is fairly flexible, with small shifts in emphasis. Common options:

• ... men hon letar ändå. (very natural)
• ... men ändå letar hon. (more emphasis on “nevertheless”)
• ... men hon letar ändå efter det. (if you add an object, ändå often sits before/near the verb or before the verb phrase)

In this sentence, putting it at the end is idiomatic and keeps the rhythm simple.

Because men connects two independent clauses (each could stand as a sentence):

• Jag tror att hon inte hittar bankkortet hemma
• hon letar ändå

In Swedish, a comma before men is common and recommended when it separates two full clauses like this.

In Swedish, tro is the usual verb for think/believe (that something is the case):

• Jag tror att... = “I think/believe that...” (opinion/assumption)

Tänka often means to think in the sense of to reflect/plan/intend:

• Jag tänker leta. = “I’m thinking of searching / I intend to search.”
  So Jag tror att... is the right choice for a belief about what’s happening.

Not normally in standard Swedish. Swedish usually requires an explicit subject in finite clauses. So you keep hon:

• ..., men hon letar ändå.

Dropping it would sound fragment-like and is generally not grammatical in neutral modern Swedish.