Learn Hardcore Polish: W naszej grupie atmosfera jest luźna, ale podczas testu wszyscy są poważni. - In our group the atmosphere is relaxed, but during the test everyone is serious.

The basic noun is grupa (group, feminine singular).
In the locative, grupa becomes grupie.
The adjective/pronoun nasza (our, feminine singular nominative) must also go to the locative to agree with grupie, so it becomes naszej.

So:

Nominative: nasza grupa (our group – subject of the sentence)
Locative: w naszej grupie (in our group – after w)

That is why you need w naszej grupie, not w nasza grupa.

Could I say atmosfera w naszej grupie jest luźna instead? Is the word order important?

Yes, you can say:

Atmosfera w naszej grupie jest luźna.

Both versions are correct:

W naszej grupie atmosfera jest luźna...
Atmosfera w naszej grupie jest luźna...

The difference is mainly in emphasis:

W naszej grupie atmosfera jest luźna
– Starts with w naszej grupie, so it emphasizes within our group (maybe in contrast to other groups).
Atmosfera w naszej grupie jest luźna
– Starts with atmosfera, so you are talking about the atmosphere first and then specifying w naszej grupie.

Grammatically, both are fine. Polish word order is relatively flexible; you mostly adjust it for emphasis and style.

What exactly does luźna atmosfera mean? Does luźna literally mean loose?

Literally, luźny means loose (not tight), but in the phrase luźna atmosfera it idiomatically means:

relaxed, easy-going, informal, not strict

So:

luźna atmosfera ≈ a relaxed / laid-back atmosphere

Other common uses of luźny:

luźne spodnie – loose trousers
luźny związek – casual relationship
luźne podejście do zasad – a relaxed/lenient attitude to rules

You could also say things like:

swobodna atmosfera – free, easy atmosphere
nieformalna atmosfera – informal atmosphere

but luźna atmosfera is very common and natural.

Why is there a comma before ale? Is it always required?

Yes, in this kind of sentence the comma before ale is required.

You have two clauses:

W naszej grupie atmosfera jest luźna
(ale) podczas testu wszyscy są poważni

They are joined by the conjunction ale (but), so Polish punctuation rules say you must put a comma before ale:

..., ale ...

So the comma in:

W naszej grupie atmosfera jest luźna, ale podczas testu wszyscy są poważni.

is obligatory.

Why is it podczas testu and not podczas test? Which case does podczas take?

The preposition podczas (during) always takes the genitive case.

The base noun is test (test, masculine).
Genitive singular of test is testu.

So:

podczas testu = during the test

Other examples:

podczas lekcji (from lekcja, genitive lekcji) – during the lesson
podczas wakacji – during the holidays
podczas rozmowy – during the conversation

So podczas testu is required; podczas test would be incorrect.

What is the difference between podczas testu, w czasie testu, and na teście?

All three can be used, but they differ slightly in nuance and frequency:

podczas testu – during the test
- Very standard, neutral. Focuses on the time period of the test.
w czasie testu – in the time of the test / during the test
- Also correct and fairly neutral. Often interchangeable with podczas testu, perhaps a bit more wordy.
na teście – literally on the test, but in Polish it means (while) at the test / in the test situation
- More about the situation or place:
  - Na teście jesteśmy bardzo skupieni. – At the test we are very focused.

In your sentence, podczas testu is the most natural choice, because you contrast the general atmosphere with how people behave during the test itself.

Why is it wszyscy są and not wszyscy jest?

Wszyscy (everyone, all of them) is plural, so the verb must also be plural:

oni są – they are
wszyscy są – everyone / all (of them) are

Jest is the third person singular form:

on / ona / ono jest – he / she / it is
atmosfera jest – the atmosphere is

So the sentence correctly contrasts:

atmosfera jest (singular)
wszyscy są (plural)

When do I use wszyscy, wszystkie, etc.? Why wszyscy here?

Wszyscy / wszystkie mean all / everyone / everything, but their form depends on gender and type of noun.

Main patterns in the nominative plural:

wszyscy – for masculine personal groups (groups that include at least one male person)
- Wszyscy przyszli. – Everyone came.
- Wszyscy uczniowie są gotowi.
wszystkie – for:
- non‑masculine personal animate (e.g. all women)
- inanimate nouns
- Wszystkie kobiety są gotowe. – All the women are ready.
- Wszystkie książki leżą na stole. – All the books are on the table.

In your sentence:

wszyscy refers to the people in the group (assumed mixed or at least one male), so wszyscy is the correct form.

If the group were definitely all women, you would normally say:

Podczas testu wszystkie są poważne.

Why is it poważni and not poważne in wszyscy są poważni?

Adjectives in Polish must agree with the noun (or pronoun) in gender, number, and case.

Here:

wszyscy is masculine personal plural (a group including at least one male).
The matching adjective form in the nominative plural is poważni.

So:

wszyscy są poważni – they (mixed / male group) are serious.

If the group were:

all women, you would say:
- wszystkie są poważne.
things / inanimate objects, you would also use:
- wszystkie są poważne.

So poważni is chosen specifically because wszyscy refers to a masculine-personal group.

Could I say jest luźno instead of jest luźna? What is the difference between luźna and luźno?

luźna is an adjective, feminine singular, matching a noun like atmosfera:
- atmosfera jest luźna – the atmosphere is relaxed.
luźno is an adverb (like loosely / in a relaxed way):
- U nas jest luźno. – With us, it’s relaxed / things are relaxed here.
- Ubrania wiszą luźno. – The clothes hang loosely.

So:

In the full sentence with the explicit subject atmosfera, you normally say:
- Atmosfera jest luźna.
If you don’t name atmosfera, you can more colloquially say:
- U nas jest luźno. – Things are relaxed here.

Your sentence deliberately mentions atmosfera, so luźna is the right choice.

Can I use a more colloquial expression, like atmosfera jest na luzie?

Yes. Na luzie is a very common colloquial phrase meaning chill / laid-back / relaxed.

Colloquial alternatives:

Atmosfera jest na luzie.
Mamy luźną atmosferę.
U nas jest na luzie.

They all sound informal, friendly, and are fully natural in spoken language.
Your original version atmosfera jest luźna is neutral and fine for both speech and writing.

Does the present tense here mean always, or just now? How does Polish express that?

In Polish, the simple present tense often covers both:

A general/habitual state
A current state

In your sentence:

W naszej grupie atmosfera jest luźna, ale podczas testu wszyscy są poważni.

This is naturally understood as a typical, repeated situation:

In our group, the atmosphere is generally relaxed, but (whenever there is) a test, everyone is serious.

Context (mention of podczas testu) makes it clear that this describes typical behavior, not just one single occasion. If you wanted to stress that something happens only sometimes, you might use:

Atmosfera w naszej grupie bywa luźna. – The atmosphere in our group is sometimes relaxed.

But in your sentence, the plain present tense is the normal way to talk about a regular pattern.