Learn Hardcore German: Auch im Winter trinken wir abends Tee im Wohnzimmer. - Even in winter we drink tea in the living room in the evenings.

Auch means also / too / even. Placing it at the beginning gives it emphasis and links to something previously said (e.g., “We drink tea in the evenings. We also do so in winter.”).
So Auch im Winter ... ≈ “Even in winter ... / In winter as well ...”
You could also place auch elsewhere depending on what you want it to modify, but here it clearly highlights im Winter.

Why is the word order Auch im Winter trinken wir ... and not ... wir trinken ...?

German follows the verb-second (V2) rule in main clauses: the finite verb (here trinken) must be in position 2.
If you put something other than the subject first (Auch im Winter), then the subject moves after the verb:

Auch im Winter (position 1)
trinken (position 2)
wir (after the verb)

This is normal and doesn’t sound “inverted” the way it might in English.

Does Auch im Winter count as one unit in the word order?

Yes. Auch im Winter functions as a single fronted adverbial phrase (a time phrase). Even though it contains two words, it occupies the first “slot” of the sentence before the verb.

What case is im Winter, and why?

im = in dem (a contraction).
dem Winter is dative, because in is a two-way preposition:

Dative for location/time (“in winter” as a time period)
Accusative for movement/direction (“into ...”)

Here it’s time/setting, not movement, so dative: im Winter.

Why is abends one word, and what does it mean?

abends is an adverb meaning in the evenings / in the evening (often in a habitual/general sense). It’s formed from Abend with an adverbial -s ending, which is common in German time expressions:

morgens (in the mornings)
mittags (at midday)
nachts (at night)
sonntags (on Sundays)

How is abends different from am Abend or abends vs am Abend?

Typical nuance:

abends = “in the evenings” (general/habitual, repeated)
am Abend = “in the evening” (often one specific evening, or a more concrete time on a particular day)

So abends trinken wir Tee suggests a regular routine.

Why is Tee not preceded by einen/den? Is that correct?

Yes. German often omits an article with beverages/food when speaking generally or about an activity:

Wir trinken Tee. = “We drink tea.” (tea in general, not a specific cup) If you want to be more specific:
Wir trinken den Tee. = “We drink the tea.” (a particular tea already known)
Wir trinken einen Tee. = “We drink a tea (a cup of tea).”

Why is it im Wohnzimmer and not in dem Wohnzimmer?

im is simply the common contraction of in dem. Both are grammatically possible, but im is much more natural in everyday German:

im Wohnzimmer = “in the living room”

You’d keep in dem mainly for emphasis/contrast or very careful style.

What case is im Wohnzimmer, and why?

It’s dative (im = in dem) because in with a location (no movement) takes dative:

Location: im Wohnzimmer (dative) = “in the living room”
Direction: ins Wohnzimmer (in das, accusative) = “into the living room”

This sentence describes where the drinking happens, not movement into the room.

Is there a preferred order for im Winter / abends / im Wohnzimmer? Could I change it?

German word order is flexible, but there are common tendencies. A common guideline is time – manner – place (often abbreviated TMP), so you might expect:

Auch im Winter trinken wir abends Tee im Wohnzimmer. (time, time-ish, place)

But you can reorder for emphasis:

... trinken wir im Wohnzimmer abends Tee. (focus more on the location first)
Abends trinken wir auch im Winter Tee im Wohnzimmer. (focus on evenings; auch shifts nuance)

The more you move things around, the more you signal what you want to emphasize.

What is the function of wir here—could it be omitted?

In German, the subject pronoun generally cannot be dropped in normal sentences (unlike Spanish/Italian).
So trinken wir needs wir explicitly. You’d only omit it in special contexts like informal notes, headlines, or commands, not in a standard statement.

Why is trinken in the present tense if the meaning is habitual?

German present tense (Präsens) commonly expresses:

actions happening now, and
habitual/general routines (“we (usually) drink ...”)

So the present tense is the normal choice for routines like drinking tea in the evenings.

Is the sentence ambiguous about “who is also drinking” vs “also in winter”?

As written, Auch im Winter strongly suggests the “also” applies to winter (“in winter as well”).
If you wanted also to apply to the people (e.g., “we, too, drink tea”), you’d more likely use:

Auch wir trinken abends Tee im Wohnzimmer. (also we / we too) German placement of auch helps clarify what it modifies.