Jag blir lycklig när vi kan vara tillsammans utan att någon är stressad.

Because bli means to become, while vara means to be.

So:

• Jag blir lycklig = I become happy / I get happy
• Jag är lycklig = I am happy

In this sentence, the idea is that being together under those conditions causes that feeling, so blir fits better. It suggests a change of state: I become happy when that happens.

När means when here.

It introduces a time clause:

• Jag blir lycklig när... = I become happy when...

It is used because the sentence is talking about a situation that happens at a certain time or under certain circumstances. In many cases, Swedish när works like English when.

In this sentence, när vi kan vara tillsammans... means when we can be together...

This is normal Swedish word order.

In a clause like this, the subject usually comes before the verb:

• vi kan vara tillsammans = we can be together

Also, after a word like när, Swedish uses subordinate clause word order, which still keeps the subject before the verb:

• när vi kan vara tillsammans
• not när kan vi vara tillsammans

Kan vi vara tillsammans? would be a question: Can we be together?

Because modal verbs in Swedish normally take the infinitive without att.

Kan is a modal verb, like can in English. After it, you use the bare infinitive:

• kan vara = can be
• vill gå = want to go
• måste läsa = must read

So vi kan vara tillsammans is correct.
You would not say vi kan att vara tillsammans.

Tillsammans means together.

So:

• vara tillsammans = be together

Depending on context, vara tillsammans can simply mean spend time together, but it can also mean be in a relationship. In this sentence, it most naturally means be together / spend time together.

Utan att means without ...ing or without the fact that..., depending on how you translate it into natural English.

Here it introduces a clause:

• utan att någon är stressad

Literally, this is something like:

• without that anyone is stressed

But natural English would usually say:

• without anyone being stressed

So utan att is a common way in Swedish to say without before a whole clause.

Examples:

• utan att säga något = without saying anything
• utan att jag visste det = without me knowing it

Någon means someone / anyone, depending on context.

Here, in utan att någon är stressad, it means anyone:

• without anyone being stressed

It refers to any person in the group or situation. The idea is that no one is feeling stressed.

It is not ingen here because the sentence is built as:

• utan att någon är stressad

rather than:

• utan att ingen är stressad

That second version would be ungrammatical in standard Swedish because ingen already has a negative meaning, and it does not fit this structure.

Here, stressad is an adjective.

• är stressad = is stressed

It describes a person's state, just like:

• är trött = is tired
• är glad = is happy

Although stressad historically comes from a past participle, in sentences like this it functions as a normal adjective meaning stressed.

Because the sentence describes a state, not a change.

• är stressad = is stressed
• blir stressad = becomes stressed / gets stressed

The sentence is talking about the condition under which the speaker is happy: when they can be together and nobody is stressed. It is not focusing on the moment someone becomes stressed.

Because jag is singular and common gender, and the basic form of the adjective is used.

• en lycklig person
• jag är lycklig

Swedish adjectives change in some contexts, but not here. For example:

• ett lyckligt barn = a happy child
• lyckliga människor = happy people

In Jag blir lycklig, the basic form lycklig is the correct one.

Mostly yes:

• Jag = I
• blir = become / get
• lycklig = happy
• när = when
• vi = we
• kan = can
• vara = be
• tillsammans = together
• utan att = without
• någon = anyone / someone
• är = is
• stressad = stressed

A very literal version would be:

• I become happy when we can be together without anyone being stressed.

That is already quite close to natural English, which is why this sentence is a good example of Swedish syntax that maps fairly well onto English.