Learn Hardcore Russian: Охрана выдала мне временный пропуск у входа. - Security issued me a temporary pass at the entrance.

Охрана is a singular noun meaning security / security staff / security service. In Russian it often works like a collective noun: grammatically singular, but referring to the group of guards/security personnel on duty.
If you want to explicitly say “the guards,” you could also say охранники (plural): Охранники выдали мне…

Why is the verb выдала in feminine form?

Past tense verbs in Russian agree in gender and number with the subject.
Охрана is grammatically feminine singular, so the past tense is feminine singular: выдала.
Compare:

Охранник выдал… (masc. sg.)
Охранники выдали… (plural)

What does the verb выдала mean, and what’s its dictionary form?

Выдала is past tense (feminine singular) of выдать (perfective), meaning to issue / hand out / give out (often officially).
In this context it’s “issued (me) a temporary pass.”

Is выдать perfective here? What would the imperfective be?

Yes, выдать is perfective, so it presents the action as completed: the pass was issued (one finished event).
The common imperfective partner is выдавать:

Охрана выдавала пропуска = “Security was issuing passes / used to issue passes” (process/repeated action).

Why is мне used (what case is it), and can it be omitted?

Мне is dative case (the indirect object), meaning to me. Russian often uses dative where English uses “to.”
It usually can’t be omitted if you want to keep “to me” in the meaning; without it, you’d just get “Security issued a temporary pass” (not saying to whom).

What case is временный пропуск, and why?

Пропуск is the direct object of выдать, so it’s in the accusative case. For an inanimate masculine noun like пропуск, accusative singular looks the same as nominative: пропуск.
Временный agrees with пропуск in gender (masc.), number (sg.), and case (acc.).

What’s the nuance of пропуск? Is it “permission,” “pass,” or “ID”?

Пропуск is typically an access pass / entry pass / badge that lets you enter a building or restricted area.
It’s not the general abstract “permission” (that’s more like разрешение). A пропуск is usually a physical document/card/badge.

What does у входа mean grammatically, and why is входа genitive?

У + genitive means by / near / at something.
So у входа = at/near the entrance.
The noun after у is genitive: вход → входа.

Could I also say на входе or входе instead of у входа?

Yes, but the nuance shifts:

у входа = physically near the entrance (very common, neutral).
на входе = at the entrance area / at the checkpoint (often where staff are stationed).
в (на) входе is not used for location like English “in the entrance”; you’d say у входа, на входе, or в вестибюле (in the lobby), depending on what you mean.

Is the word order fixed? Could I rearrange it?

Russian word order is flexible and changes emphasis. Neutral is close to what you have:
Охрана выдала мне временный пропуск у входа.
Other natural options:

У входа охрана выдала мне временный пропуск. (focus: location first)
Мне у входа выдали временный пропуск. (focus: “to me”; subject can be omitted if obvious)

Why is there no word for “the/a” in Russian, and how do I know if it’s “a pass” or “the pass”?

Russian has no articles, so пропуск can be “a pass” or “the pass” depending on context.
Here, without additional context, learners often translate it as a temporary pass, because it sounds like a new item being issued. If both speakers already know which pass, it could be the temporary pass.

Can I replace охрана with something more specific, and would the verb change?

Yes:

Сотрудник охраны выдал мне временный пропуск… = “A security employee issued me…”
Охранник выдал мне… (masc. sg. → выдал)
Охранница выдала мне… (fem. sg. → выдала)
Охранники выдали мне… (plural → выдали)