Learn Hardcore Dutch: Het is te laat om ons nog aan te melden; de lijst is vol. - It is too late to sign up; the list is full.

Het is a dummy subject used in impersonal statements with adjectives like te laat, moeilijk, mogelijk, etc. It corresponds to English “It is …”.
Er introduces existence (“there is/are”), which doesn’t fit here. You’re not asserting something exists; you’re evaluating a situation: “It is too late …”

What does the te in te laat mean, and is it the same te as in om … te?

They are different uses of the same word:

In te laat, te means “too” (excessive degree): “too late.”
In om … te …, te is the infinitive marker “to” (as in “to sign up”).

Is om required in om ons … aan te melden?

After adjectives like te laat, Dutch strongly prefers (and many guides effectively require) om before a te-infinitive. So Het is te laat om ons … aan te melden is the natural form.
Without om (e.g., Het is te laat ons … aan te melden) sounds odd or stilted.

Why is it ons and not wij/we?

Wij/we is a subject pronoun. Here, the verb is reflexive: zich aanmelden (“to register oneself”). You need the object/reflexive form: ons (“ourselves”).
Reflexive forms:

1st sg: me
2nd sg: je
3rd sg: zich
1st pl: ons
2nd pl: je/jullie
3rd pl: zich

Do I have to include the reflexive pronoun? Could I say just om nog aan te melden?

If you mean “for us to sign up,” you must include ons. Without it, aanmelden could mean “register (someone/something),” and the sentence becomes incomplete or changes meaning.
For oneself: Ik wil me aanmelden, We willen ons aanmelden.

Why is it aan te melden and not te aanmelden?

Aanmelden is a separable verb: main-clause finite form splits (e.g., Ik meld me aan), but in a te-infinitive the particle and stem surround te: aan te melden.
Pattern:

Main clause: Ik meld me aan.
Te-infinitive: … om me aan te melden.
Putting te before the whole verb (te aanmelden) is incorrect.

Where does nog go, and can I move it?

Standard placement is after the object pronoun and before the verb cluster: om ons nog aan te melden.
Alternatives like om nog ons aan te melden or om ons aan te melden nog are unnatural.
Meaning here: nog = “still/any longer.”

Could I express this with a negative instead of nog?

Yes. A very common alternative is:

We kunnen ons niet meer aanmelden; de lijst is vol.
Here niet meer = “no longer/anymore.” It’s often more straightforward than “te laat om … nog …”

What’s the difference between aanmelden, inschrijven, and opgeven?

(Zich) aanmelden: to sign up/register, often the act of presenting yourself (e.g., online form, front desk). You can add a preposition: aanmelden voor (an event), aanmelden bij (an organization/desk).
(Zich) inschrijven: to enroll/register (often more formal or administrative, e.g., courses, municipalities). Typically inschrijven voor.
(Zich) opgeven: to put one’s name down/volunteer/sign up (colloquial); also means “to give up” in a different context, so mind ambiguity.
All three can overlap in casual use, but nuance varies by context.

Why is it de lijst and not het lijst?

Because lijst is a common-gender noun in Dutch, so it takes de. You just have to memorize noun genders; there’s no rule that would make this het.

Is vol the right word here? How is it different from volledig?

Yes. Vol means “full/no more space.”
Volledig means “complete/entire,” not “full.”
Regional note: Belgian Dutch often uses volzet for “full (no places left).”

Can I join the two parts with a conjunction instead of a semicolon?

Absolutely:

Het is te laat om ons nog aan te melden, want de lijst is vol. (coordinating, no word order change)
Het is te laat om ons nog aan te melden, omdat de lijst vol is. (subordinating, verb to the end: is)

Why a semicolon here? Would a comma or period work?

A semicolon links two closely related independent clauses without a conjunction. A period would also be fine. A plain comma between two independents is less standard in careful writing unless you add want/omdat.

Can I say We zijn te laat om ons nog aan te melden instead of starting with Het is?

Yes. Both are idiomatic:

Het is te laat om ons nog aan te melden.
We zijn te laat om ons nog aan te melden.
The first is impersonal; the second puts the focus on “we.”

Could I replace nog with something like meer on its own?

Not by itself. Meer alone means “more.” The negative pair is niet meer (“no longer/anymore”).
Positive nog = “still/yet”: ons nog aanmelden.
Negative: ons niet meer aanmelden.

What prepositions go with aanmelden?

aanmelden voor + activiteit/evenement: We willen ons aanmelden voor de cursus.
aanmelden bij + organisatie/balie/website: Meld je aan bij de receptie.
In your sentence, the object (what for) is implied by de lijst being full.