Learn Hardcore Norwegian: Gruppechatten vår er vanligvis hyggelig, men i går ble en kommentar litt for privat. - Our group chat is usually pleasant, but yesterday a comment became a bit too private.

So definite noun + possessive is the standard neutral way: gruppechatten vår.
If you put it before (vår gruppechat), the noun becomes indefinite and the phrase often sounds a bit more emphatic or contrastive, like our group chat (as opposed to someone else’s).

Why is it gruppechatten and not just gruppechat?

Gruppechatten is the definite form of (en) gruppechat:

en gruppechat = a group chat
gruppechatten = the group chat

In this sentence you are talking about a specific, known chat that everyone in the conversation already knows about, so the definite form is used.
Then the possessive vår is added after it: gruppechatten vår = our group chat.

What exactly is the difference between er and ble here?

er is the present tense of å være (to be), used for a general, ongoing state:
- Gruppechatten vår er vanligvis hyggelig = It is usually nice (as a general fact).
ble is the past tense of å bli (to become), often used for a change of state:
- i går ble en kommentar litt for privat = yesterday a comment became / got a bit too private.

So er hyggelig describes a usual or typical quality, while ble … privat describes something turning private at a particular time.

Could I say “men i går var en kommentar litt for privat” instead of “ble”?

You can say var, and it is grammatically correct, but it shifts the nuance:

ble privat = became / turned private (focus on the change that happened yesterday)
var privat = was private (it simply was that way yesterday; no strong sense of change)

In context, ble is more natural because it suggests that the comment crossed a line at some point: it got a bit too private.

What does vanligvis mean exactly, and where can it go in the sentence?

Vanligvis means usually / generally / most of the time.

Its position in the sentence can vary a bit, but common placements are:

Gruppechatten vår er vanligvis hyggelig.
Vanligvis er gruppechatten vår hyggelig.

Putting it after the verb (er vanligvis hyggelig) is very typical and neutral.
Putting it at the beginning (Vanligvis er …) puts more emphasis on “usually” as a contrast to what happened yesterday.

What is the nuance of hyggelig here? Is it like “nice”, “fun”, or “cozy”?

Hyggelig is broad and context-dependent. In this sentence it means something like:

pleasant
friendly
nice to be in

It does not necessarily mean cozy in the physical sense (like a cozy room). It rather describes the atmosphere in the group chat: people are kind, the tone is friendly, and it feels comfortable to participate.

What is the difference between hyggelig and koselig?

Both can sometimes be translated as nice or cozy, but:

hyggelig is more general: nice, pleasant, friendly.
- Det var en hyggelig samtale. = It was a nice/pleasant conversation.
koselig often has a stronger sense of cozy, warm, intimate, homely.
- Vi hadde en koselig kveld hjemme. = We had a cozy evening at home.

In the context of a group chat, hyggelig is more common and natural; koselig would sound more like the chat feels warm and cozy, which is possible but a bit more specific in tone.

How should I understand litt for privat?

Litt for privat literally means a little too private:

litt = a little, somewhat
for (before an adjective) = too (excessively)
privat = private (also used about personal things)

So litt for privat expresses a gentle criticism: not horribly inappropriate, but it crossed the line a bit.
It sounds slightly softer and less harsh than just for privat (too private).

Why is it en kommentar and not kommentaren?

En kommentar is the indefinite form: a comment.
You’re introducing this comment for the first time; it hasn’t been specified before, so the indefinite article is used.

If you said kommentaren, that would mean the comment, implying that both speaker and listener already know exactly which comment is being referred to, from earlier context.

What genders do gruppechat and kommentar have, and how can I tell?

Both are masculine nouns in Bokmål:

(en) gruppechat – gruppechatten
(en) kommentar – kommentaren

There is no fully reliable way to guess gender from the form alone; you usually have to memorize it.
However, many loanwords and modern technical words, like chat, kommentar, tend to be treated as masculine in everyday Bokmål, especially with the article en.