Learn Hardcore Esperanto: Mi petas, ke vi venu frue morgaŭ kaj proponu novajn ideojn. - I ask that you come early tomorrow and propose new ideas.

The ending -u is the volitive (often called the subjunctive/imperative form) in Esperanto.
After verbs like peti (to ask/request), deziri (to wish), ordoni (to order), proponi (to suggest), you normally use ke + verb in -u to express what you want someone to do:

Mi petas, ke vi venu… = I request that you come…
Li ordonis, ke ili foriru. = He ordered that they leave.

If you said Mi petas, ke vi venas, it would sound more like “I request that you are coming,” describing a fact rather than making a request; it is not idiomatic Esperanto in this context.

Why is there a comma after Mi petas before ke?

In Esperanto, you normally put a comma before ke when it introduces a full subordinate clause:

Mi pensas, ke…
Ŝi diris, ke…
Mi petas, ke…

So Mi petas, ke vi venu… follows that general punctuation rule.
In very short, tightly bound expressions some writers might omit the comma, but with Mi petas, ke… the comma is standard and recommended.

Could I say Mi petas vi venu frue morgaŭ without ke?

No. That is not correct Esperanto.
The options are:

Use ke
- -u:
  - Mi petas, ke vi venu frue morgaŭ.
Or use an infinitive construction:
- Mi petas vin veni frue morgaŭ.

But you cannot simply drop ke and keep vi venu; ke is the conjunction that links the main verb (petas) to the subordinate clause (vi venu…).

What is the difference between Mi petas, ke vi venu… and Mi petas vin veni…?

Both are grammatically correct and very common:

Mi petas, ke vi venu frue morgaŭ…
Literally: I request that you come early tomorrow…
Here the thing requested is the whole clause after ke.
Mi petas vin veni frue morgaŭ…
Literally: I request you to come early tomorrow…
Here vin is the direct object of petas, and veni is an infinitive complement.

They are practically equivalent in meaning; the ke + -u pattern is slightly more explicit about the requested action as a clause, but in everyday speech both forms are fine.

Could I say Mi petas, ke vi venos frue morgaŭ with venos instead of venu?

Not for the same meaning.
Venos is future indicative and talks about what will happen:

Mi scias, ke vi venos frue morgaŭ. = I know that you will come early tomorrow.

With peti, you are expressing a wish/request, so you normally use -u:

Mi petas, ke vi venu frue morgaŭ. = I ask that you (please) come early tomorrow.

Using venos after Mi petas, ke… would sound like you are stating or predicting a future fact, not politely requesting it.

Why is it frue, not frua?

Frue is an adverb, describing how/when you come; it modifies the verb venu:

veni frue = to come early.

Frua is an adjective and must describe a noun:

frua mateno = an early morning.
frua renkontiĝo = an early meeting.

In Mi petas, ke vi venu frue morgaŭ, you are not saying “a tomorrow that is early”; you are saying “that you come early (in relation to tomorrow)”, so the adverb frue is the correct choice.

Is frue morgaŭ the only correct order, or can I say morgaŭ frue?

You can say frue morgaŭ or morgaŭ frue; both are correct and natural.
They both mean “early tomorrow”, and in most contexts there is no real difference in meaning. Esperanto word order is fairly flexible with time and manner adverbs, so speakers freely use both:

Mi petas, ke vi venu frue morgaŭ.
Mi petas, ke vi venu morgaŭ frue.

Why do both novajn and ideojn have -n at the end?

The -n is the accusative ending, used mainly for direct objects and to show direction.
In proponu novajn ideojn:

The direct object of proponu is ideojn (“ideas”), so it gets -n.
Adjectives must agree with their noun in number and case, so nova also becomes novajn (plural accusative) to match ideojn.

Pattern:

singular nominative: nova ideo
plural nominative: novaj ideoj
singular accusative: novan ideon
plural accusative: novajn ideojn

Could I say novan ideon instead of novajn ideojn?

Yes. That would mean “a new idea” (one idea) instead of “new ideas” (more than one).
Esperanto has no indefinite article (a/an), so:

novan ideon = a new idea
novajn ideojn = new ideas

Which one you use depends purely on whether you want to talk about one idea or several.

Why is there no word for “please”, like “bonvolu”, in this sentence? Could I use it?

The sentence expresses politeness with Mi petas… (“I request / I ask (you)”), so a separate “please” is not required.
You could rephrase it with bonvolu:

Bonvolu veni frue morgaŭ kaj proponi novajn ideojn.

Bonvolu is literally “be so kind as to…”, and is a very common way to say “please do X”.
Both Mi petas, ke vi venu… and Bonvolu veni… are polite; the style is slightly different, but the politeness level is similar.

Can I move morgaŭ somewhere else in the sentence, like at the beginning?

Yes. Time expressions are quite free in Esperanto. For example, all of these are acceptable:

Morgaŭ mi petas, ke vi venu frue kaj proponu novajn ideojn.
Mi morgaŭ petas, ke vi venu frue kaj proponu novajn ideojn. (less usual, but possible in context)
Mi petas, ke vi morgaŭ venu frue kaj proponu novajn ideojn.

Changing the position can slightly change the emphasis, but not the basic meaning. The original word order is very natural and neutral.