Learn Hardcore Portuguese (Portugal): O treinador disse que qualquer campeã também passa por muitas competições e treinos difíceis. - The coach said that any champion also goes through many competitions and hard training sessions.

If you drop the article (Treinador disse…), it sounds incomplete or like a headline, not normal speech.
You only normally omit the article when you’re speaking in general about a profession:

Treinadores ganham pouco. – Coaches earn little. (coaches in general)

What exactly does qualquer mean in qualquer campeã, and why does it go before the noun?

Qualquer means “any” in a non‑specific, open way:

qualquer campeã – any champion (whoever she may be)

It always comes before the noun:

qualquer dia – any day
qualquer pessoa – any person

If you put it after the noun (campeã qualquer), it sounds informal and a bit dismissive, more like:

uma campeã qualquer – just some champion / some random champion

So qualquer campeã here is neutral and general, not dismissive.

Why is it campeã and not campeão? What’s the rule for this change?

Campeão is the masculine form (“male champion”), and campeã is the feminine (“female champion”).

Pattern: many nouns ending in -ão form the feminine in -ã:

campeão → campeã
alemão → alemã (German person)
cristão → cristã (Christian)

So qualquer campeã is talking explicitly about a female champion.
If you wanted it to be neutral or masculine, you’d say qualquer campeão.

Why is the verb passa (present) after disse que (past)? Would passou also be possible?

Disse is past (he/she said), but passa is present: goes through / has to go through.

Portuguese often keeps the present in reported speech when the statement is:

a general truth
a habitual action

So:

O treinador disse que qualquer campeã também passa por…
→ The coach said that any champion (in general) goes through many competitions…

If you say:

…disse que qualquer campeã também passou por…

then you’re shifting the focus to past champions, to what happened in the past, not to a general rule about champions. Both can be correct, but the meaning changes slightly:

passa – timeless, general rule
passou – past, specific situation/time period

What does passar por mean here? Why not just passar or passar em?

In this context, passar por means “to go through / to undergo / to experience”:

passar por muitas competições – to go through many competitions
passar por momentos difíceis – to go through hard times

Without a preposition, passar usually means “to pass” in a more literal or different sense (to pass by, to pass an exam, to pass something to someone).

Passar em is used for things like passing an exam:

passar no exame / passar em medicina – to pass the exam / to get into medicine

So for the idea “to go through difficult things”, passar por is the natural choice.

Why is it qualquer campeã também passa and not também qualquer campeã passa or passa também? Where can também go?

Também means “also / too / as well”. The most neutral and common position is before the main verb:

qualquer campeã também passa…

Other positions are possible, but they change focus or sound more marked:

Também qualquer campeã passa… – emphasizing any champion too (more stylistic/marked)
qualquer campeã passa também… – can emphasize “also” at the end; sounds a bit heavier or more spoken‑style

In normal, neutral sentences, learn this pattern:

subject + também
- verb
  → qualquer campeã também passa…

Does muitas in muitas competições e treinos difíceis apply to both competições and treinos, or only to competições?

Grammatically, muitas directly modifies only competições:

muitas competições
treinos difíceis (with no explicit muitos)

In real speech, listeners will usually understand that there are many competitions and many difficult training sessions. However, if you want to be completely clear and symmetrical, you can say:

…por muitas competições e muitos treinos difíceis.

Both versions are used. The original is natural and idiomatic, especially in spoken language, but the repeated muitos avoids any ambiguity.

Why is the adjective after the noun in treinos difíceis and not difíceis treinos?

In Portuguese, the default position for most adjectives is after the noun:

treinos difíceis – difficult training sessions
casas grandes – big houses
competição importante – important competition

Putting the adjective before the noun is possible but usually:

sounds more literary or expressive, or
changes the nuance slightly (more subjective/emotional).

So difíceis treinos is possible but would be unusual in everyday speech and would sound stylistically marked. In standard, neutral Portuguese, use noun + adjective: treinos difíceis.

Why is it treinos difíceis and not treinos difícil? How does adjective agreement work here?

Adjectives in Portuguese must agree with the noun in:

number (singular/plural)
gender (masculine/feminine, when relevant)

Here:

treinos – masculine plural
difícil → plural form difíceis

So we get:

treino difícil – one difficult training session
treinos difíceis – difficult training sessions (plural)

That’s why treinos difícil (plural noun + singular adjective) would be incorrect.

What is the role of que in disse que qualquer campeã…? Could it be left out?

Que here is a subordinating conjunction that introduces reported speech / a subordinate clause:

O treinador disse que… – The coach said that…

In standard Portuguese, you cannot omit this que:

❌ O treinador disse qualquer campeã também passa… (incorrect)
✅ O treinador disse que qualquer campeã também passa… (correct)

So think of disse que as “said that” – they go together when you introduce what was said.