Jeg lager en omelett til frokost når jeg har tid.

Yes. Å lage often means to make or to prepare, and it is very common for food.

So Jeg lager en omelett means I make / prepare an omelet.

For an English speaker, it helps to remember:

• lage mat = make/cook food
• lage en omelett = make an omelet

You would not normally use gjøre here. Gjøre is more like do in general expressions, not make food.

Because omelett is a countable noun, and in Norwegian singular countable nouns usually need an article when they are indefinite.

So:

• en omelett = an omelet
• omeletten = the omelet

Since the sentence is talking about an omelet in a general, indefinite way, en is needed.

In Norwegian, til frokost is the natural expression for for breakfast / as breakfast.

So:

• til frokost = for breakfast
• til lunsj = for lunch
• til middag = for dinner

English uses for, but Norwegian usually uses til in this kind of phrase.

Because meal words like frokost, lunsj, and middag often appear without an article when you are talking about them in a general sense.

So Norwegian says:

• til frokost
• til lunsj
• til middag

This works a bit like English expressions such as for breakfast, where you also do not say for a breakfast.

Norwegian present tense is used very broadly. It can describe:

• habits
• general truths
• repeated actions
• sometimes future meaning, depending on context

In this sentence, Jeg lager en omelett til frokost når jeg har tid sounds like a habitual or whenever I have time statement.

So the present tense is exactly what you would expect here:

• lager = make / am making / will make, depending on context
• har = have

Here it is best understood as a repeated action: I make an omelet for breakfast when I have time.

Når is used for when in the sense of:

• whenever
• repeated situations
• present/future-type situations

So here:

• når jeg har tid = when / whenever I have time

By contrast, da is usually used for one specific time in the past.

Compare:

• Når jeg har tid, lager jeg en omelett.
  = When/whenever I have time, I make an omelet.
• Da jeg hadde tid i går, lagde jeg en omelett.
  = When I had time yesterday, I made an omelet.

So når is the right choice in your sentence.

Because når jeg har tid is a full subordinate clause, and Norwegian normally includes the subject there just as English does.

The sentence has two clauses:

1. Jeg lager en omelett til frokost
2. når jeg har tid

The second clause still needs its own subject:

• jeg har tid = I have time

So it would be ungrammatical to leave out jeg there.

Yes, it is a very common expression.

• å ha tid = to have time

So:

• jeg har tid = I have time
• jeg har ikke tid = I don’t have time

This works very much like English. There is no article before tid here.

Yes, absolutely.

You can say:

• Jeg lager en omelett til frokost når jeg har tid.
• Når jeg har tid, lager jeg en omelett til frokost.

Both are natural.

But if you move Når jeg har tid to the front, the word order in the main clause changes. Norwegian follows the V2 rule, so the verb comes before the subject in the main clause:

• Når jeg har tid, lager jeg ...

Not:

• Når jeg har tid, jeg lager ...

That last version is wrong.

Yes, omelett is the standard Norwegian spelling.

It corresponds to English omelet / omelette. English has more than one common spelling, but Norwegian normally uses omelett.

Also note that it is a common-gender noun, so:

• en omelett
• omeletten

By itself, it most naturally sounds like a habit or a general pattern:

• I make an omelet for breakfast when I have time.

In other words: Whenever I have time, that is what I do.

If you wanted a more specific one-time future meaning, context would usually make that clear, or you might add a time expression such as:

• I morgen lager jeg en omelett til frokost hvis jeg har tid.
  = Tomorrow I’ll make an omelet for breakfast if I have time.

So your original sentence is best understood as general/habitual.

Yes, you could say hvis jeg har tid, but the nuance changes slightly.

• når jeg har tid = when/whenever I have time
• hvis jeg har tid = if I have time

Når sounds more like this happens whenever the condition is met.
Hvis sounds more conditional or uncertain.

So:

• Jeg lager en omelett til frokost når jeg har tid
  = whenever I have time, I make one
• Jeg lager en omelett til frokost hvis jeg har tid
  = I make one if I happen to have time

Both are possible, but når fits the idea of a repeated habit especially well.