Se mangio troppo velocemente, mi fa male lo stomaco.

In Italian, a real or general if sentence often uses the present tense in both parts:

• Se mangio troppo velocemente, mi fa male lo stomaco.

This means something like If I eat too fast, my stomach hurts / I get a stomachache.

Italian does not normally use the conditional after se in this kind of sentence. So se mangerei would be wrong here.

Italian usually leaves out subject pronouns when they are already clear from the verb ending.

• mangio = I eat

So Se mangio... already means If I eat...
You could say Se io mangio..., but that would add emphasis, for example to contrast with someone else.

Mangio is the first-person singular present indicative of mangiare.

• mangiare = to eat
• mangio = I eat

So in this sentence, mangio matches the implied subject I.

Velocemente is an adverb, meaning quickly / fast. It describes how you eat.

• mangio = I eat
• velocemente = quickly
• troppo velocemente = too quickly / too fast

Here, troppo modifies the adverb velocemente.

Troppo velocemente is correct and understandable. But in everyday speech, many Italians would more often say:

• Se mangio troppo in fretta, mi fa male lo stomaco.

In fretta is a very common way to say quickly / in a hurry.
So both are fine, but troppo in fretta may sound a bit more conversational.

Italian often expresses pain with the pattern:

• mi fa male + body part

Literally, this is something like it causes pain to me, but the natural meaning is simply:

• my ... hurts
• I have pain in my ...

So:

• mi fa male lo stomaco = my stomach hurts

This is a very common structure in Italian.

Because the verb agrees with lo stomaco, which is the grammatical subject of the expression.

• lo stomaco = singular
• therefore fa = singular

So the structure is basically:

• lo stomaco fa male a me
• more natural Italian: mi fa male lo stomaco

The mi means to me, but it is not the subject.

Because stomaco is a masculine singular noun beginning with s + consonant.
In Italian, those nouns usually take lo in the singular definite article:

• lo stomaco
• lo studente
• lo sport

So il stomaco is incorrect.

With body parts, Italian very often uses the definite article instead of a possessive, especially when the owner is already clear.

So Italian prefers:

• mi fa male lo stomaco

rather than:

• mi fa male il mio stomaco

The possessive is usually unnecessary because mi already shows whose stomach it is.

Mi is an indirect object pronoun meaning to me.

So:

• mi fa male lo stomaco = the stomach hurts to me
  natural English: my stomach hurts

This pattern is very common with body parts and physical sensations:

• Mi fa male la testa = My head hurts
• Mi fanno male i piedi = My feet hurt

Yes, the word order can change, but the most natural everyday version is:

• Mi fa male lo stomaco

You may also hear:

• Lo stomaco mi fa male

That version can sound more marked or emphatic, because it puts extra focus on lo stomaco.

Because the se clause comes first:

• Se mangio troppo velocemente, mi fa male lo stomaco.

When the conditional clause comes before the main clause, Italian often uses a comma, just as English does. It helps separate the two parts clearly.

If the order is reversed, the comma is often omitted:

• Mi fa male lo stomaco se mangio troppo velocemente.

Yes. That is another very common way to express a stomachache.

• Mi fa male lo stomaco = My stomach hurts
• Ho mal di stomaco = I have a stomachache

They are close in meaning, but not identical in structure.
The sentence you were given focuses on the pain happening when you eat too fast.