Learn Hardcore Russian: Очередь у кассы тянется до входа. - The line at the cashier stretches to the entrance.

Because the preposition у (meaning by / near / at) normally requires the genitive case.
So у кассы = near the cash register / at the checkout.
Base form: касса (nominative) → genitive singular: кассы.

Why is вход in the form входа?

Because the preposition до (meaning up to / as far as) also requires the genitive case.
So до входа = up to the entrance.
Base form: вход (nominative) → genitive singular: входа.

What does у кассы mean exactly, and how is it different from в кассе or к кассе?

у кассы = by/next to the cash register (location near something)
в кассе = in the cash register / in the cashier’s office (inside; usually not what you want here)
к кассе = toward the cash register (direction; movement destination)

So for a line that is physically located near the register, у кассы is the natural choice.

What part of speech is тянется, and why does it end in -ется?

тянется is a verb: 3rd person singular, present tense of тянуться.
The ending -ется matches the subject очередь (feminine singular), so it means it stretches / it extends.

Why is the verb reflexive (тянуться)—what does -ся add?

In Russian, тянуть is usually to pull / to stretch something (transitive), while тянуться is to stretch / to extend (intransitive)—the subject “does it itself,” so to speak.
Here, the line isn’t pulling something else; it extends, so тянется fits.

Is тянется describing a physical “stretching,” or can it be used figuratively?

Both are possible in Russian, but in this exact context it’s a very common literal spatial description: the line’s length reaches a point.
It can also be figurative in other contexts (for example, time or processes “drag on”), but here it’s about distance.

Why is очередь in the nominative (очередь) and not some other case?

Because очередь is the grammatical subject of the sentence—the thing that тянется.
So it stays in nominative: Очередь … тянется …

Does до входа mean “all the way to the entrance” or just “toward the entrance”?

до + genitive normally means up to / as far as a boundary point, often implying it reaches that point.
If you only mean “in the direction of,” Russian typically uses something like к входу (toward the entrance) instead.

Is the word order fixed? Could I say До входа тянется очередь у кассы?

Russian word order is flexible. You can reorder for emphasis:

Очередь у кассы тянется до входа. (neutral)
До входа тянется очередь у кассы. (emphasizes how far it goes)
У кассы очередь тянется до входа. (emphasizes location near the register)

The cases keep the meaning clear even when the order changes.

How would the sentence change if there are multiple checkouts?

Common options:

Очередь у касс тянется до входа. = A line by the checkouts stretches to the entrance. (касс = genitive plural after у)
Or you might specify: Очередь к одной из касс… if you mean a particular one.

What’s the pronunciation/stress of the key words here?

очередь: stress on the second syllable → очерЕдь
кассы: stress on the first syllable → кАссы
тянется: stress on the first syllable → тЯнется
входа: stress on the first syllable → вхОда