Jag blir lugn av frisk luft.

Because bli / blir means become, while är means am/is/are.

So:

• Jag är lugn = I am calm
• Jag blir lugn = I become calm / I get calm

In this sentence, the idea is that fresh air makes the speaker calm, so Swedish uses blir.

Lugn is an adjective, and its form has to match the noun or pronoun it describes.

Here it describes jag, which refers to a person. With common-gender words and people in the singular, the basic form is used:

• en lugn person = a calm person
• Jag är lugn = I am calm

You get lugnt with a neuter noun or in some adverb-like uses:

• ett lugnt barn = a calm child
• Det är lugnt = It is calm / No problem

So Jag blir lugn is the correct form.

Here av means something like from, because of, or through the effect of.

So Jag blir lugn av frisk luft means:

• I become calm from fresh air
• I become calm because of fresh air
• Fresh air has a calming effect on me

This is a very common use of av in Swedish when something causes a feeling or reaction:

• Jag blir trött av värmen = I get tired from the heat
• Hon blev glad av nyheten = She became happy because of the news

In Swedish, general or uncountable things often appear without an article.

Luft is usually treated as an uncountable noun, like air in English. So when speaking generally, Swedish says:

• frisk luft = fresh air
• varm luft = warm air
• kall luft = cold air

This is similar to English, where you normally say fresh air, not a fresh air.

Because luft is an en-word in Swedish: en luft.

Adjectives agree with the noun:

• en noun → basic adjective form: frisk
• ett noun → -t form: friskt

So:

• frisk luft = fresh air
• friskt vatten = fresh water

That is why frisk is correct here.

Yes, frisk can mean different things depending on context.

Common meanings include:

• healthy / well
  
  • Jag är frisk nu = I am well now
• fresh
  
  • frisk luft = fresh air
• sometimes brisk / cool / invigorating depending on context

So in frisk luft, it definitely means fresh air, not healthy air in a literal word-for-word sense.

Yes, it is natural and understandable.

It sounds like a general statement about how fresh air affects you. Depending on context, Swedish speakers might also say:

• Frisk luft gör mig lugn. = Fresh air makes me calm.
• Jag känner mig lugn av frisk luft. = I feel calm from fresh air.
• Jag blir lugn när jag får frisk luft. = I become calm when I get fresh air.

Your original sentence is perfectly good Swedish.

No, those would not be correct.

• Jag blir lugnt av frisk luft is wrong because lugnt does not match jag here.
• Jag känner lugn av frisk luft is also wrong, because with känna in this meaning Swedish usually says känner mig lugn.

Correct alternatives are:

• Jag blir lugn av frisk luft
• Jag känner mig lugn av frisk luft

So if you use bli, keep lugn. If you use känna, usually use mig as well: känner mig lugn.

Blir is the present tense of bli.

• bli = to become
• blir = become / becomes / am becoming / get

Examples:

• Jag blir lugn = I become calm / I get calm
• Jag blev lugn = I became calm
• Jag har blivit lugn = I have become calm

So in your sentence, the speaker is expressing a present, general fact: fresh air makes them calm.

Yes, but the basic order here is the most neutral:

• Jag blir lugn av frisk luft.

You can move parts for emphasis, but then Swedish follows its normal word-order rules. For example:

• Av frisk luft blir jag lugn.
  This is possible, but more marked. It puts emphasis on av frisk luft.

Notice that when something other than the subject comes first, the verb still stays in second position:

• Av frisk luft blir jag lugn
• not Av frisk luft jag blir lugn

That is a standard Swedish word-order pattern.

A careful approximate pronunciation is:

yah bleer luŋn ahv frisk lʉft

A few useful notes:

• Jag is often pronounced more like ya in everyday speech.
• blir has a long i sound.
• lugn can be tricky; the g is not pronounced as a hard g.
• frisk has a clear sk sound here, like sk in skin? Not exactly. In standard Swedish, it is more like a sharp s+k cluster.
• luft has the Swedish u, which does not sound like English oo.

You do not need perfect pronunciation at first, but blir, lugn, and luft are good words to listen to in native audio.