Нам нужны подсказка и ключ-карта.

Нам is the dative form of мы (we).

So:

• мы = we
• нам = to us / for us

In this pattern, Russian often says something more like To us, X is needed instead of using a direct equivalent of English we need X.

So the structure is:

• Нам = to us
• нужны = are needed
• подсказка и ключ-карта = a hint and a key card

Very literally: To us are needed a hint and a key card.
Natural English: We need a hint and a key card.

Because подсказка и ключ-карта is a compound subject: two separate things joined by и (and).

Even though each noun is singular by itself:

• подсказка = hint
• ключ-карта = key card

together they make a plural idea:

• подсказка и ключ-карта = a hint and a key card

So the predicate must also be plural:

• нужны = needed (plural)

Compare:

• Нам нужна подсказка. = We need a hint.
• Нам нужна ключ-карта. = We need a key card.
• Нам нужны подсказка и ключ-карта. = We need a hint and a key card.

Because Russian is not using a normal transitive verb here.

English says:

• We need X
  where X is the direct object.

Russian often uses a different construction with нужен / нужна / нужно / нужны, which is not behaving like a normal verb to need. Instead, it works more like:

• X is needed to someone

That is why:

• the person who needs something is in the dative: нам
• the thing needed is in the nominative: подсказка, ключ-карта

So in this sentence, подсказка и ключ-карта are grammatically more like the subject than the object.

Нужны is the short-form plural of the adjective нужный (necessary / needed).

Forms:

• нужен — masculine singular
• нужна — feminine singular
• нужно — neuter singular
• нужны — plural

In sentences like this, these short forms are used predicatively, meaning they act as the main predicate:

• Мне нужен ключ. = I need a key.
• Мне нужна карта. = I need a map.
• Мне нужны ключи. = I need keys.

So although English translates this with the verb need, Russian is literally using something closer to is needed / are needed.

Yes—in a way.

In the present tense, Russian normally omits the verb to be.

So Russian says:

• Нам нужны подсказка и ключ-карта.

not:

• Нам есть нужны... ❌

The idea of are is understood automatically.

This is very common in Russian:

• Он врач. = He is a doctor.
• Мы дома. = We are at home.
• Мне нужна помощь. = I need help.

So there is no extra present-tense to be word in the sentence.

Yes. Russian word order is fairly flexible because the cases show the grammatical roles.

This sentence could also appear as:

• Подсказка и ключ-карта нам нужны.
• Нужны нам подсказка и ключ-карта.

These all mean roughly the same thing, but the emphasis changes.

A rough guide:

• Нам нужны подсказка и ключ-карта.
  Neutral: We need a hint and a key card.

• Подсказка и ключ-карта нам нужны.
  Emphasizes the things needed: It’s a hint and a key card that we need.

So the original sentence is perfectly normal, but not the only possible order.

Ключ-карта is a compound noun meaning key card.

It is written with a hyphen because Russian often writes some compound borrowed nouns this way.

It is treated as feminine because the second part, карта, is feminine. That is why, if it were the only thing needed, you would say:

• Нам нужна ключ-карта.

In this sentence it stays in the nominative singular because it is one of the things that are needed.

A useful point: in compounds like this, the second part often determines the gender and usually carries the ending changes.

You do not know from the sentence alone, because Russian has no articles like a/an/the.

So:

• подсказка can mean a hint or the hint
• ключ-карта can mean a key card or the key card

The context tells you which meaning is intended.

For example:

• If the speaker is talking generally, it may mean a hint and a key card.
• If both speaker and listener already know which hint and which key card, it may mean the hint and the key card.

This is completely normal in Russian.

You can use that as a learning tool, but not as a natural English translation.

It helps you understand the grammar:

• Нам = to us
• нужны = are needed

But in natural English, you should usually translate it as:

• We need a hint and a key card.

So the literal translation is useful for grammar, while the natural translation is better for actual meaning.

Usually, with a noun phrase naming the thing needed, нужен / нужна / нужны is the most straightforward choice.

So:

• Нам нужны подсказка и ключ-карта. = We need a hint and a key card.

Надо and нужно are very often used with an infinitive:

• Нам нужно найти подсказку и ключ-карту. = We need to find a hint and a key card.
• Нам надо открыть дверь. = We need to open the door.

Using нужны here focuses on the objects needed.
Using нужно / надо more often focuses on what needs to be done.

So in this exact sentence, нужны is the best fit.