Αν και είμαι κουρασμένος, θέλω να μαγειρέψω κάτι απλό και να μην παραγγείλω.

No. Αν και is a fixed expression meaning although or even though.

Even though αν by itself often means if, in αν και the two words work together as one connector:

• Αν και είμαι κουρασμένος = Although I am tired

So you should learn αν και as a set phrase, not as if + and.

Greek often expresses states like this with:

• είμαι = I am
• an adjective = tired

So:

• είμαι κουρασμένος = I am tired
  literally, I am tired in the same basic way as English.

The adjective changes according to gender and number:

• masculine: κουρασμένος
• feminine: κουρασμένη
• neuter: κουρασμένο

So a female speaker would usually say:

• Αν και είμαι κουρασμένη...

Κουρασμένος is an adjective agreeing with the speaker.

In this sentence it is:

• masculine
• singular
• nominative

That is because the speaker is I, and the adjective matches the person speaking.

Common forms:

• κουρασμένος = tired (masculine)
• κουρασμένη = tired (feminine)
• κουρασμένο = tired (neuter)

This is a very common feature of Greek: adjectives must agree with the noun or pronoun they describe.

Modern Greek does not use an infinitive the way English does.
Instead, after verbs like θέλω (I want), Greek normally uses να + a finite verb form.

So:

• θέλω να μαγειρέψω = I want to cook
• literally something like I want that I cook, though you should not translate it word-for-word.

This pattern is extremely common:

• θέλω να φύγω = I want to leave
• μπορώ να έρθω = I can come
• πρέπει να διαβάσω = I must study

Μαγειρέψω here is the aorist subjunctive form used after να. In practical terms, it often refers to the action as a single whole event.

Compare:

• να μαγειρέψω = to cook / to cook once / to get some cooking done
• να μαγειρεύω = to be cooking / to cook regularly / to cook as an ongoing activity

In this sentence, the speaker means one complete action:

• I want to cook something simple

So the aorist form μαγειρέψω is the natural choice.

In standard Greek, the second να is normally repeated:

• θέλω να μαγειρέψω ... και να μην παραγγείλω

That is because both actions depend on θέλω:

• I want to cook
• and (I want) not to order

Repeating να makes the structure clear and natural.
So the standard pattern is:

• θέλω να X και να Y

not usually:

• θέλω να X και Y

This is a very common learner question.

Greek usually uses:

• δεν for ordinary statement negation
• μη(ν) with να, commands, and certain non-indicative structures

So:

• δεν παραγγέλνω = I am not ordering / I do not order
• να μην παραγγείλω = not to order / that I not order

Because this sentence has να + verb, the correct negative is μην:

• να μην παραγγείλω

Here παραγγείλω means to order, especially in the sense of ordering food, often from a restaurant or delivery place.

So:

• να μην παραγγείλω = not to order (food)

Greek often leaves the object understood from context.
In English we might say:

• not order in
• not order food
• not get takeout / delivery

All of those fit the idea here.

It can, but it does not have to.

Greek often omits things that are obvious from context. In a sentence about cooking, παραγγείλω naturally suggests ordering food.

So this is perfectly normal:

• θέλω να μαγειρέψω ... και να μην παραγγείλω

If you wanted to be more explicit, you could say:

• να μην παραγγείλω φαγητό = not to order food

Because κάτι behaves like a neuter singular idea, the adjective after it is usually also neuter singular.

So:

• κάτι = something
• απλό = simple (neuter singular)

Together:

• κάτι απλό = something simple

This is very common:

• κάτι καλό = something good
• κάτι ωραίο = something nice
• κάτι μικρό = something small

With κάτι, the most normal order is:

• κάτι + adjective

So:

• κάτι απλό
• κάτι ωραίο
• κάτι ενδιαφέρον

This is the standard and natural order here.
Putting the adjective before κάτι would not sound natural in this kind of sentence.

Θέλω is present tense: I want.

Greek uses the present here because the speaker is expressing a current इच्छा/desire at the moment of speaking:

• Although I’m tired, I want to cook something simple...

That is exactly parallel to English I want.

Yes, Greek word order is fairly flexible, though some versions sound more neutral than others.

The given sentence:

• Αν και είμαι κουρασμένος, θέλω να μαγειρέψω κάτι απλό και να μην παραγγείλω.

is very natural and neutral.

You could also say:

• Θέλω να μαγειρέψω κάτι απλό και να μην παραγγείλω, αν και είμαι κουρασμένος.

That still makes sense, but it slightly changes the flow and emphasis.
Putting Αν και είμαι κουρασμένος first sets up the contrast immediately.

Yes, it is standard to place a comma after the introductory αν και clause here:

• Αν και είμαι κουρασμένος, θέλω...

It works much like English:

• Although I am tired, I want...

The comma marks the end of the subordinate clause and makes the sentence easier to read.

Only the adjective would normally change:

• Αν και είμαι κουρασμένη, θέλω να μαγειρέψω κάτι απλό και να μην παραγγείλω.

Everything else stays the same.

Then Greek would more likely use the present subjunctive form:

• θέλω να μαγειρεύω

But in this sentence that would sound less natural, because κάτι απλό points to one specific cooking action.
So να μαγειρέψω is the better choice for:

• cook something simple

The contrast is roughly:

• να μαγειρέψω = cook, get the cooking done
• να μαγειρεύω = be cooking / cook regularly / cook as an ongoing process