Learn Hardcore Chinese: wǒ xiāngxìn nǐ huì chénggōng.

会 here mainly marks a future likelihood or expected result, similar to English “will” or “be going to”.
It implies: given what I know now, I expect you to succeed.
Besides future, 会 can also mean “to be able to (learned skill)”, but in this sentence it’s the future meaning.

Can I omit 会 and just say 我相信你成功?

You normally cannot say 我相信你成功 in this context; it sounds incomplete or wrong.
To talk about a future result, you need something like:

我相信你会成功。 – I believe you will succeed.
我相信你一定会成功。 – I believe you will definitely succeed.

Without 会 (or another marker like 一定, 能够, etc.), the sentence doesn’t sound natural for this meaning.

Is 成功 (chénggōng) a verb or a noun here?

In this sentence, 成功 functions as a verb meaning “to succeed”.
So the basic structure is:

我 (subject) – 相信 (verb) – 你会成功 (object clause “that you will succeed”).

成功 can also be a noun (“success”), but here it’s used as “to succeed”, not “success” as a thing.

Why is there no word like “to” (as in “to succeed”) in Chinese?

Chinese doesn’t use a separate word like English “to” before verbs in this way.
The idea of “to + verb” is just expressed by the verb itself:

你会成功 = “you will succeed” (not “you will to succeed”).
So 成功 already covers “to succeed” without needing an extra marker.

Why don’t we use 是 (shì) like 我是相信你会成功?

是 is generally used to link nouns or noun phrases (A is B), not a subject directly to a verb phrase.
You can’t say 我是相信你会成功 as a normal equivalent of “I believe you will succeed”; it sounds wrong or at least very odd.
The correct pattern is simply:

我相信…… – I believe that …

是 is not needed before another main verb like 相信.

What’s the difference between 我相信你 and 我相信你会成功?

我相信你 = “I trust you / I believe in you” in a general sense (your character, abilities, honesty).
我相信你会成功 = “I believe you will succeed”, focusing on a specific outcome in the future.

You can also say both together for emphasis:
我相信你，我相信你会成功。 – I believe in you; I believe you will succeed.

Is there a more direct way to say “I believe in you” in Chinese using this sentence?

我相信你 by itself is already a common and natural way to say “I believe in you”.
If you want to connect it more explicitly to success, you can say:

我相信你，一定会成功。 – I believe in you; (you) will definitely succeed.
我相信你会成功。 – I believe you will succeed.

For strong encouragement, many speakers add 一定 (“definitely”).

Could I use 能 (néng) or 可以 (kěyǐ) instead of 会 here?

You could say 我相信你能成功, and it’s acceptable, but the nuance is slightly different:

会成功 – more about a future outcome / likelihood: “will succeed.”
能成功 – more about ability or possibility: “can manage to succeed,” “are capable of succeeding.”

可以成功 is less common and sounds a bit more like “it is possible for you to succeed,” not as natural in encouragement as 会成功 or 能成功.
For the standard encouraging sentence, 会成功 is most natural.

How would I say this sentence as a question: “Do you believe I will succeed?”

You can simply reverse the roles and add 吗:

你相信我会成功吗？ – Do you believe I will succeed?

Structure: 你 (you) + 相信 (believe) + 我会成功 (I will succeed) + 吗 (question particle).

How do I negate this sentence to say “I don’t believe you will succeed”?

You place 不 before 相信:

我不相信你会成功。 – I don’t believe you will succeed.

The pattern is: 我 + 不 + 相信 + (clause).
You do not negate 会 here; you negate the act of believing.

What’s the difference between 相信, 觉得, and 认为? They all look like “think/believe”.

In this context:

相信 – “to believe / have faith in / trust”, often with a feeling of confidence or trust.
觉得 – “to feel / to think” in a subjective, casual way (often about impressions or feelings).
认为 – “to consider / to hold the opinion that”, more formal or logical.

For encouraging someone about their success, 相信 is the most natural choice:
我相信你会成功。 sounds warmer and more supportive than 我觉得你会成功 or 我认为你会成功.

Can I add 一定 (yídìng) or 肯定 (kěndìng) for emphasis, and where do they go?

Yes, both are very common to emphasize certainty and encouragement. Typical positions:

我相信你一定会成功。 – I believe you will definitely succeed.
我相信你肯定会成功。 – I believe you will surely / definitely succeed.

They go before 会, modifying the degree of certainty about the future event 成功.

Any pronunciation or tone tips for 相信 (xiāngxìn) and 成功 (chénggōng)?

相 (xiāng) – 1st tone (high and level). Don’t let it fall.
信 (xìn) – 4th tone (sharp falling).
Together: 相信 – high level + sharp fall.
成 (chéng) – officially 2nd tone, but often sounds like a light rising from mid to high.
功 (gōng) – 1st tone (high and level).

Practice the rhythm of the whole sentence:
wǒ xiāngxìn nǐ huì chénggōng – try to keep the tones clear and the sentence smooth and even.