Learn Hardcore Russian: Я забыл пропуск дома, поэтому мне пришлось ждать у входа. - I forgot my pass at home, so I had to wait by the entrance.

Забыл is the past tense of забыть (perfective). Perfective is natural because it presents the forgetting as a single completed event: I forgot (and as a result I didn’t have it).
If you used the imperfective забывал, it would usually mean I used to forget / I was in the habit of forgetting / I forgot (repeatedly), which doesn’t fit this one-time situation as well.

What exactly does пропуск mean in Russian?

Пропуск here means a pass/badge/entry permit (e.g., an office badge, student pass, security pass). In many contexts it’s the physical card you need to enter a building.

Why is it пропуск and not свой пропуск (my pass)?

Russian often omits possessives like мой/свой when ownership is obvious from context. Since the subject is я, it’s naturally understood as my pass.
You can add it for emphasis/contrast:

Я забыл свой пропуск дома = I forgot my pass (specifically my own, not someone else’s).

Why is дома used instead of something like в доме?

Дома is an adverb meaning at home / at the house (location). It’s the standard, idiomatic way to say at home in Russian.
В доме literally means in the house (inside the building) and is used when you mean the physical interior, not the general concept of “home.”

Is the meaning “I forgot it at home” or “I left it at home”? Why does Russian use забыл?

Russian забыть + object + place commonly covers the idea of forgetting something somewhere, which in English often comes out as left it somewhere.
So Я забыл пропуск дома is best understood as I left my pass at home / I forgot my pass at home. Both are natural English translations depending on style.

Why is there a comma before поэтому?

Because this is a compound sentence with two clauses:
1) Я забыл пропуск дома
2) мне пришлось ждать у входа
поэтому (therefore/so) connects them, and Russian normally uses a comma before поэтому when it links two independent clauses.

What does мне пришлось literally mean, and why is it in the dative (мне)?

пришлось is the past form of прийтись in the meaning to have to / to be forced to / to end up having to.
It commonly uses the dative for the person affected: мне пришлось = I had to (it fell to me / it ended up that I had to).
This is an impersonal-style construction: the “necessity” is the main idea, not the person as an active doer.

Why is it пришлось ждать (with an infinitive)? What governs ждать?

пришлось is often followed by an infinitive to say what you had to do:

мне пришлось + infinitive = I had to + verb
So мне пришлось ждать = I had to wait.

Why is ждать used without an object? Wouldn’t it need “wait for…”?

In Russian, ждать can be used absolutely (without stating what you’re waiting for) when it’s understood from context: I had to wait (there).
If you want to specify, you add an object (often in the genitive/accusative depending on meaning and style):

ждать друга = wait for a friend
ждать автобуса = wait for the bus

Why is it у входа and not у вход or входа without у?

у is a preposition meaning by/near/at (the side of) and it requires the genitive case.

вход (nominative) → входа (genitive)
So у входа = by the entrance / at the entrance.

Could I also say на входе? What’s the difference from у входа?

Yes, на входе is possible and often means at the entrance area / at the entry point, sometimes implying “at the checkpoint/doorway” as a functional location.
у входа is more simply near/by the entrance (physical proximity).
Both can translate as at the entrance, but у входа is the safer default for “near the entrance.”

What is the neutral word order here, and can it be changed?

The sentence uses a very neutral, natural order: cause → therefore → result. You can reorder for emphasis, but meaning stays basically the same:

Поскольку я забыл пропуск дома, мне пришлось ждать у входа. (more formal: since)
Я забыл дома пропуск, поэтому мне пришлось ждать у входа. (emphasizes at home)
Мне пришлось ждать у входа, потому что я забыл пропуск дома. (result first, then cause)

How is this sentence pronounced/stressed (roughly)?

Approximate stress (capitalized stressed vowel):
Я забЫл прОпуск домА, поэтОму мнЕ пришлОсь ждАть у вхОдa.
Notes:

поэтому stress is typically поэтОму.
пришлось stress is пришлОсь.
входа is вхОдa.