Learn Hardcore German: Ich schicke den Zählerstand noch am selben Tag an die Hausverwaltung. - I send the meter reading to the property management the very same day.

der Zählerstand = nominative (dictionary form / subject)
den Zählerstand = accusative (direct object)
dem Zählerstand = dative (would be used if it were an indirect object, but here it isn’t)

So: Ich schicke den Zählerstand … = I send the meter reading …

Why does the sentence use an die Hausverwaltung—what case is die Hausverwaltung in here?

Here an is a two-way preposition (can take accusative or dative). With schicken, it typically expresses a direction/target, so it uses the accusative:

an die Hausverwaltung (accusative) = to the property management (as a destination)
If it were describing a static location (not a destination), you’d expect dative, e.g. an der Tür = at the door.

Could I also say zur Hausverwaltung or an der Hausverwaltung?

zur Hausverwaltung (= zu der) is often possible and common: Ich schicke den Zählerstand zur Hausverwaltung. It sounds like “to the property management” in a general sense.
an der Hausverwaltung is generally wrong here, because that would mean something like “at the property management” (a location), not a recipient.

So for sending something: an die Hausverwaltung or zur Hausverwaltung are the typical options.

What does noch mean in this sentence, and where does it usually go?

noch here means something like still / yet / as early as / even depending on context; in this sentence it implies by that same day / on the same day (still)—i.e., it will happen without delay.

Position: noch often sits near the time phrase it modifies:

… noch am selben Tag … (very natural)
You could also move it slightly, but the given placement clearly ties it to am selben Tag.

What is the grammar of am selben Tag?

am is a contraction of an dem (dative):

an dem selben Tag → am selben Tag
German commonly uses an + dative with days to mean on that day.

Also note the adjective ending: selben is weakly declined because it follows dem (a definite article in dative masculine/neuter).

Why is it selben and not gleichen? Are they interchangeable?

They’re close, and both are common:

am selben Tag = on that very/same day (often used when referencing a day already mentioned or clearly implied)
am gleichen Tag = on the same day (also fine; sometimes feels a bit more neutral/explicit)

In many contexts they are interchangeable, but am selben Tag is especially idiomatic in administrative-style German.

What exactly does Zählerstand mean, and how is the word built?

Zählerstand is a compound noun:

Zähler = meter/counter (e.g., electricity meter, water meter)
Stand = “state/level/reading” (like “the current value/position”)

So Zählerstand = meter reading (the number shown on the meter).

Why is the verb schicke in second position, and what’s the basic word order here?

German main clauses follow the V2 rule: the finite verb (here schicke) is in position 2.

Structure here is straightforward:

Ich (subject)
schicke (finite verb)
Then:
- den Zählerstand (direct object)
- noch am selben Tag (time information)
- an die Hausverwaltung (recipient/target)

Can I move parts of the sentence around (e.g., put the time first)?

Yes. You can front other elements for emphasis; the verb still stays second:

Noch am selben Tag schicke ich den Zählerstand an die Hausverwaltung. (emphasizes “the same day”)
An die Hausverwaltung schicke ich den Zählerstand noch am selben Tag. (emphasizes the recipient)

When you move something to the front, Ich often moves behind the verb: … schicke ich …

Is Hausverwaltung always feminine, and what does it refer to in practice?

Yes, die Hausverwaltung is feminine (noun ending -ung is almost always feminine). In practice it refers to the entity/person managing an apartment building—property management, building management, or the management company handling tenants, utilities, repairs, etc.

How would I say this more formally or more informally?

More formal (common in emails/letters):

Ich übermittle den Zählerstand noch am selben Tag an die Hausverwaltung.
Ich werde den Zählerstand noch am selben Tag an die Hausverwaltung senden.

More informal:

Ich schicke den Zählerstand noch am selben Tag zur Hausverwaltung.
Ich schick den Zählerstand noch am selben Tag an die Hausverwaltung. (dropping the -e in speech)