Learn Hardcore Portuguese (Brazil): Eu espero que meu pai chegue cedo para jantar com a família. - I hope that my father arrives early to have dinner with the family.

In Portuguese, verbs that express desire, hope, doubt, fear, etc. (like esperar, querer, tomara, duvidar) are usually followed by que + subjunctive:

Eu espero que ele chegue cedo.
Eu quero que você venha.
Tomara que chova.

So espero que meu pai chegue is grammatically required; espero que meu pai chega is incorrect.

What exactly is the form chegue? How is it formed from chegar?

Chegue is:

Mood: subjunctive
Tense: present subjunctive
Person: 1st and 3rd person singular (it serves both eu and ele/ela/você)

Present subjunctive of chegar:

que eu chegue
que tu chegues
que ele / ela / você chegue
que nós cheguemos
que vós chegueis (rare in Brazil)
que eles / elas / vocês cheguem

Formation rule (regular -ar verbs):
Take the eu form of the present indicative (eu chego), drop the -o, and add:

-e, -es, -e, -emos, -eis, -em

So: chego → cheg- → chegue, chegues, chegue, cheguemos, chegueis, cheguem.

Why does espero que need the subjunctive, when in English we just say I hope my dad arrives?

Portuguese is stricter about marking uncertainty, desire, and subjectivity with a different verb mood.

English: I hope (that) my dad arrives early.
Uses normal present tense, but the idea of hope is carried by hope itself.
Portuguese:
- Eu espero que meu pai chegue cedo.
  The verb esperar already expresses hope, and then Portuguese uses the subjunctive (chegue) to show that the action is not a fact yet, just wished for.

Other similar patterns:

Eu duvido que ele venha. (I doubt he will come.)
Eu quero que você entenda. (I want you to understand.)

In all of these, English keeps an indicative form (comes, understand), while Portuguese switches to the subjunctive (venha, entenda).

Can I drop Eu and just say Espero que meu pai chegue cedo…?

Yes, absolutely.

Portuguese usually drops subject pronouns when the verb ending already shows who the subject is. Both are correct:

Eu espero que meu pai chegue cedo.
Espero que meu pai chegue cedo.

This sentence is clear without Eu, because espero (first person singular) already tells you the subject is eu.

Why is it para jantar com a família and not something like para o jantar com a família?

Both patterns are possible, but they mean slightly different things.

para jantar com a família
- jantar here is a verb in the infinitive: to have dinner.
- This expresses purpose:
  - Ele chega cedo para jantar.
    He arrives early in order to have dinner.
para o jantar com a família
- o jantar here is a noun: the dinner.
- This refers to a specific event or meal:
  - Ele chega cedo para o jantar.
    He arrives early for the dinner (the event itself).

In everyday conversation, para jantar com a família (with the verb) is more natural to say why he is arriving early.

Why do we say com a família and not just com família?

In Portuguese, family nouns usually take the definite article when referring to the person’s family as a known group:

com a família = with the family (the speaker’s or someone specific’s family)
com minha família = with my family
com a minha família = also correct (and very common in Brazil)

Without the article:

com família sounds incomplete or very unusual. It would only make sense in special contexts, like a short label:
Homens com família pagam menos impostos.
(Men with families pay less tax.)

In your sentence com a família is the natural way to say with the family.

In European Portuguese I often see o meu pai. Why is it meu pai here, without the article?

This is mostly a Brazil vs Portugal preference.

Brazilian Portuguese:
With singular close family members, Brazilians often drop the article and say:
- meu pai, minha mãe, meu irmão, minha filha, etc.
  o meu pai is not wrong, but meu pai is more typical.
European Portuguese:
It is more common to include the article:
- o meu pai, a minha mãe, o meu irmão…

So in Brazil:

Eu espero que meu pai chegue cedo. (most natural)
Eu espero que o meu pai chegue cedo. (possible, a bit more formal or emphatic)

Why is the preposition para used here, and not por?

Because para expresses purpose or goal, while por tends to express cause, means, duration, movement through, etc.

In para jantar com a família, the idea is:

He arrives early in order to have dinner with the family.
→ purpose/goal → para

If you used por instead:

chegar cedo por jantar com a família
This is not idiomatic and sounds wrong, because por does not naturally introduce a purpose with an infinitive like that.

Simple contrast:

Eu estudo para aprender. (I study in order to learn.)
Eu estudo por prazer. (I study because of pleasure / for pleasure.)

Where can cedo go in the sentence? Is chegue cedo the only option?

Cedo (early) is pretty flexible, but not all positions sound equally natural.

Most natural placements here:

Eu espero que meu pai chegue cedo para jantar com a família.
Eu espero que meu pai, para jantar com a família, chegue cedo. (more formal, with commas)

You can also say:

Eu espero que meu pai chegue para jantar cedo com a família.
Grammatically possible, but a bit less clear; it may sound like having dinner early rather than arriving early.

By default, adverbs like cedo most commonly appear right after the verb they modify:

chegar cedo (arrive early)
sair tarde (leave late)
comer rápido (eat quickly)

Can I say mais cedo instead of cedo? Is there a difference?

Yes, you can, and there is a small difference:

cedo = early (no comparison)
- Eu espero que meu pai chegue cedo.
  I hope my dad arrives early.
mais cedo = earlier, earlier than usual / earlier than some reference point
- Eu espero que meu pai chegue mais cedo.
  I hope my dad arrives earlier (than usual / than he normally does / than others).

In your sentence:

…chegue cedo para jantar… → just says “early enough to have dinner.”
…chegue mais cedo para jantar… → suggests “earlier than he normally arrives,” so that this time he can have dinner with the family.