Wenn ich vegetarisch koche, mache ich oft einen Salat mit Sellerie und Lauch.

Because wenn introduces a subordinate clause in German. In subordinate clauses, the conjugated verb usually goes to the end.

So:

• Wenn ich vegetarisch koche = when/if I cook vegetarian food

Compare:

• Main clause: Ich koche vegetarisch.
• Subordinate clause: ..., wenn ich vegetarisch koche.

That final verb position is one of the most important German word order patterns to learn.

Because the sentence begins with a subordinate clause:

• Wenn ich vegetarisch koche, ...

When that whole clause comes first, it takes up the first position in the sentence. In German main clauses, the conjugated verb must come second, so the next part becomes:

• mache ich oft einen Salat ...

Structure:

• [Wenn ich vegetarisch koche] [mache] [ich] [oft einen Salat ...]

This is a standard verb-second pattern.

It can mean when or if, depending on context.

In this sentence, it most naturally means something like:

• Whenever I cook vegetarian food, I often make a salad...

So it expresses a repeated situation or habit.

A rough guide:

• wenn = when for repeated events, or if
• als = when for a single event in the past

Examples:

• Wenn ich Zeit habe, lese ich. = When/If I have time, I read.
• Als ich ein Kind war, ... = When I was a child, ...

Because vegetarisch here is being used adverbially, not as an adjective directly before a noun.

It describes how someone cooks:

• Ich koche vegetarisch. = I cook vegetarian food / I cook in a vegetarian way.

If it came before a noun, then it would need adjective endings:

• Ich koche vegetarische Gerichte. = I cook vegetarian dishes.

So:

• vegetarisch = adverb-like use, no ending
• vegetarische Gerichte = adjective before a plural noun, so it gets an ending

Einen Salat is accusative, because it is the direct object of machen.

You are making what?

• einen Salat

Since Salat is masculine (der Salat), its accusative form is:

• der Salat → nominative
• den/einen Salat → accusative

Here the sentence uses the indefinite article:

• einen Salat

Because Salat is a masculine noun, and masculine nouns change in the accusative.

With ein-words, the masculine forms are:

• Nominative: ein Salat
• Accusative: einen Salat

So:

• Ein Salat ist gesund. = A salad is healthy.
  Here Salat is the subject.
• Ich mache einen Salat. = I make a salad.
  Here Salat is the direct object.

Yes, mit always takes the dative.

The reason you do not see a special dative ending here is that Sellerie and Lauch are used without articles in a general, ingredient-like sense.

So:

• mit Sellerie und Lauch = with celery and leek

If you added articles, then you would clearly see the dative:

• mit dem Sellerie und dem Lauch

But that sounds more specific, as if you mean particular celery and particular leek. For ingredients in a general sense, German often leaves the article out.

Because in ingredient lists or general food descriptions, German often uses nouns without articles.

So:

• mit Sellerie und Lauch
• mit Tomaten
• mit Reis
• mit Käse

This is similar to English, where we also often say:

• with celery and leek
• with rice
• with cheese

If you want to refer to specific items, you can use articles:

• mit dem Lauch aus dem Garten
• mit dem Sellerie, den ich gestern gekauft habe

It means to cook vegetarian food or to cook in a vegetarian way.

It does not necessarily focus on the speaker personally being a vegetarian. It focuses on the type of cooking.

So:

• Ich koche vegetarisch. = I cook vegetarian food.

A person can say this even if they are not always vegetarian; it may just describe what they are cooking on that occasion or as a habit.

Oft means often. It is an adverb of frequency.

In this sentence it comes after the verb:

• mache ich oft einen Salat

That is a very natural position in German.

German adverbs can move around somewhat, but this placement is common and neutral. It corresponds well to English:

• I often make a salad

Yes, some parts could be rearranged, but the original sentence is the most natural neutral version.

For example, you could also say:

• Ich mache oft einen Salat mit Sellerie und Lauch, wenn ich vegetarisch koche.

That puts the wenn clause at the end instead of the beginning.

What you cannot do is ignore the normal verb rules. For example, after a fronted subordinate clause, the main clause must still have the verb in second position:

• Correct: Wenn ich vegetarisch koche, mache ich oft ...
• Incorrect: Wenn ich vegetarisch koche, ich mache oft ...

Usually yes, but there is a small nuance.

In everyday German, Sellerie can refer generally to celery, but in some contexts it may be ambiguous because German distinguishes:

• Stangensellerie = stalk celery / celery
• Knollensellerie = celeriac / celery root

So in a cooking sentence like this, Sellerie is often understood broadly from context. If you want to be very precise, you can use Stangensellerie.

Lauch usually means leek.

In some contexts it can also be a more general word related to the allium family, but in normal food vocabulary, the usual meaning is leek.

So:

• Sellerie und Lauch = celery and leek

Because it is the first word of the sentence.

In German, all nouns are capitalized, but the first word of a sentence is also capitalized no matter what kind of word it is.

So here:

• Wenn is capitalized because it starts the sentence.
• Salat, Sellerie, and Lauch are capitalized because they are nouns.

Yes. That is correct, but it is slightly different in style.

• Ich koche vegetarisch. = I cook vegetarian food / I cook in a vegetarian way.
• Ich koche vegetarische Gerichte. = I cook vegetarian dishes.

The first is more idiomatic and general. The second is a bit more explicit because it names the noun Gerichte (dishes).

Both are good German.