Dass die Nachhilfe geholfen hat, gibt ihr neues Selbstvertrauen.

Yes. In German, an entire dass-clause can act as the subject of a sentence.

• Dass die Nachhilfe geholfen hat = That the tutoring helped
• This whole clause is the subject of the main verb gibt.

So the structure is:

• Subject: Dass die Nachhilfe geholfen hat
• Verb: gibt
• Indirect object: ihr
• Direct object: neues Selbstvertrauen

In English we often rephrase with it as a dummy subject:

• Dass die Nachhilfe geholfen hat, gibt ihr neues Selbstvertrauen.
  → The fact that the tutoring helped gives her new self-confidence.
  → It gives her new self-confidence that the tutoring helped. (more literal)

Because dass introduces a subordinate clause, and in German subordinate clauses the finite verb goes to the very end.

• Main clause word order: Die Nachhilfe hat geholfen. (verb in 2nd position)
• Subordinate clause with dass: …, dass die Nachhilfe geholfen hat. (finite verb at the end)

So:

• geholfen is the past participle.
• hat is the finite auxiliary verb, so it must come last in the subordinate clause.

Nachhilfe is a noun meaning tutoring, private lessons, or extra help (usually academic).

• Nachhilfe in Mathe bekommen = to get extra tuition in math
• Nachhilfelehrer = private tutor

So die Nachhilfe here is the tutoring / extra lessons she received.

The verb geben (to give) usually has:

• A dative indirect object: to whom?
• An accusative direct object: what?

In this sentence:

• gibt = gives
• ihr = to her (dative)
• neues Selbstvertrauen = new self-confidence (accusative)

That’s why it’s ihr (dative singular of sie = she), not sie:

• sie (nominative) = she
• ihr (dative) = to her

So literally:
Dass die Nachhilfe geholfen hat, gibt ihr neues Selbstvertrauen.
→ That the tutoring helped gives *to her new self-confidence.*

Neues Selbstvertrauen is the direct object of gibt, so it is in the accusative case.

Selbstvertrauen is:

• a neuter noun: das Selbstvertrauen
• here: accusative singular neuter

When a neuter noun in the accusative does not have an article (das, ein, etc.), the adjective takes a strong ending:

• Nominative neuter: neues Selbstvertrauen
• Accusative neuter: neues Selbstvertrauen

So neues here = new (accusative singular neuter, strong ending -es).

Yes. Selbstvertrauen is a noun, so it is always capitalized in German.

Meaning:

• Selbst = self
• Vertrauen = trust

Together: self-confidence, self-assurance, literally self-trust.

The tenses reflect a time relation:

• geholfen hat (present perfect) expresses a completed action in the past:
  The tutoring has helped.
• gibt (present) expresses a current, ongoing result:
  …gives her new self-confidence (now).

So the idea is:

• Because the tutoring has helped (completed), it now gives her new confidence (present result).

Dass with double s is a subordinating conjunction meaning that (introducing a clause):

• Ich weiß, dass die Nachhilfe geholfen hat.
  = I know that the tutoring helped.

Das with one s can be:

• the article das (the, neuter), or
• the demonstrative pronoun das (that, this, it).

Example:

• Das hat ihr geholfen. = That/This helped her.
• Das Selbstvertrauen = the self-confidence

In your sentence, it must be dass (with ss) because it introduces the clause dass die Nachhilfe geholfen hat.

In German, subordinate clauses are separated from the main clause with a comma.

• Dass die Nachhilfe geholfen hat, = subordinate clause (subject)
• gibt ihr neues Selbstvertrauen. = main clause

Rule: when a sentence starts with a subordinate clause, you must place a comma before the main clause:

• Wenn es regnet, bleibe ich zu Hause.
• Weil ich müde bin, gehe ich früh ins Bett.
• Dass die Nachhilfe geholfen hat, gibt ihr neues Selbstvertrauen.

Yes, you can express a very similar idea with a more straightforward subject:

• Die Nachhilfe hat ihr neues Selbstvertrauen gegeben.
  = The tutoring has given her new self-confidence.

This version:

• makes Die Nachhilfe the subject of the sentence
• uses the present perfect hat gegeben instead of gibt
• avoids using a dass-clause as the subject

The original sentence, however, focuses more on the fact that the tutoring helped as the thing that gives her confidence.

Yes, that is a grammatical but slightly different sentence:

• Weil die Nachhilfe geholfen hat, hat sie neues Selbstvertrauen.
  = Because the tutoring helped, she has new self-confidence.

Differences:

• Weil gives a direct cause: because it helped, she has confidence.

• Dass in the original sentence treats the fact that it helped as a thing that gives confidence:

  • Dass die Nachhilfe geholfen hat, gibt ihr neues Selbstvertrauen.
    = The fact that the tutoring helped gives her new self-confidence.

So:

• weil-sentence: focuses on reason
• dass-sentence: focuses on the fact itself as a source of confidence

The word order is not absolutely fixed, but there is a strong preference.

Most natural:

• …, gibt ihr neues Selbstvertrauen. (dative ihr before accusative neues Selbstvertrauen)

You could say:

• …, gibt neues Selbstvertrauen ihr.

but this sounds unusual or poetic and would normally require a special context or emphasis (for example, stressing ihr heavily).

Standard, neutral German normally places:

• dative pronoun before the accusative object:
  gibt ihr neues Selbstvertrauen.