Learn Hardcore Greek: Όποτε έχω λίγο χρόνο, μαγειρεύω στο σπίτι αντί να τρώω έξω. - Whenever I have a little time, I cook at home instead of eating out.

This is also correct and natural. The nuance is small; όποτε makes the idea of every time / whenever a bit stronger and more clearly habitual.

Why is there no word for I in έχω λίγο χρόνο and μαγειρεύω? How do I know it means I?

In Greek, subject pronouns (εγώ, εσύ, αυτός etc.) are usually dropped, because the verb ending shows the person:

έχω = I have
έχεις = you (singular) have
μαγειρεύω = I cook
μαγειρεύεις = you cook

So:

(Εγώ) έχω λίγο χρόνο = I have a little time
(Εγώ) μαγειρεύω = I cook

You only add εγώ for emphasis or contrast, e.g. Εγώ μαγειρεύω, όχι εσύ.

What exactly does λίγο add to έχω λίγο χρόνο? How is it different from just έχω χρόνο?

έχω χρόνο = I have time (enough time; time is available)
έχω λίγο χρόνο = I have a little time / some time (limited amount)

λίγο softens the idea and suggests the time is not a lot, just a small amount. It matches the English nuance: whenever I have a bit of time rather than whenever I have time in general.

Why is μαγειρεύω in the present tense and not μαγειρέψω?

Greek has two basic aspects for verbs:

Present stem (e.g. μαγειρεύω) → ongoing, repeated, habitual actions.
Aorist stem (e.g. μαγειρέψω) → single, complete actions (especially in the subjunctive or past).

In your sentence:

μαγειρεύω στο σπίτι = I cook at home (as a habit, regularly)

This matches the meaning: whenever I have a bit of time, I (usually) cook at home.

If you used μαγειρέψω, it would need να (e.g. να μαγειρέψω) and would talk about a single event or a specific intention, not a general habit, so it wouldn’t fit here.

Why is it αντί να τρώω and not just αντί τρώω?

The structure is:

αντί να
- subjunctive (present or aorist verb)

So you must have να:

αντί να τρώω έξω = instead of eating out
αντί να φάω έξω = instead of eating out (once, as a single event)

Without να, αντί τρώω is ungrammatical.

αντί by itself normally takes a noun / pronoun:

αντί για καφέ = instead of coffee
αντί για μένα = instead of me

What is the difference between αντί να τρώω έξω and αντί να φάω έξω?

Both mean instead of eating out, but the aspect is different:

αντί να τρώω έξω (present subjunctive) → ongoing / repeated action:
instead of (usually / regularly) eating out
αντί να φάω έξω (aorist subjunctive) → one complete action:
instead of eating out (this one time)

In your sentence, the whole thing is about a habit (whenever I have time, I cook at home rather than go out to eat regularly), so αντί να τρώω έξω is the natural choice.

What does στο σπίτι literally mean, and why do we need στο here?

στο = σε + το:

σε = in / at / to
το = the (neuter singular article)

So στο σπίτι literally is in/at the house.

Here, μαγειρεύω στο σπίτι = I cook at home.

You normally need σε (combined as στο, στη, στους etc.) with places:

μένω στο σπίτι = I live at home
δουλεύω στο γραφείο = I work at the office

Could I say just μαγειρεύω σπίτι without στο? Would that be wrong?

You can hear μαγειρεύω σπίτι in casual speech, and people will understand it. However:

μαγειρεύω στο σπίτι is the standard, fully correct form.
Omitting στο sounds more colloquial or telegraphic, and in some contexts a bit incomplete.

So for learners and for neutral speech, prefer μαγειρεύω στο σπίτι.

What part of speech is έξω, and where can it appear in the sentence?

έξω is an adverb meaning outside / out.

In this sentence, it modifies τρώω:

τρώω έξω = I eat out (literally: I eat outside)

You can move έξω a bit, but the most natural are:

αντί να τρώω έξω (as in the original)
αντί να τρώω έξω στο εστιατόριο (adding more detail)

Putting έξω far away from τρώω usually sounds awkward.

Is the comma after χρόνο necessary? Could I say Μαγειρεύω στο σπίτι όποτε έχω λίγο χρόνο instead?

You can place the subordinate clause either first or second:

Όποτε έχω λίγο χρόνο, μαγειρεύω στο σπίτι…
Μαγειρεύω στο σπίτι, όποτε έχω λίγο χρόνο…

In Greek, when the subordinate clause comes first, you normally use a comma after it. When it comes second, the comma is optional and depends on rhythm and emphasis. Both of these are fine:

Μαγειρεύω στο σπίτι όποτε έχω λίγο χρόνο. (no comma)
Μαγειρεύω στο σπίτι, όποτε έχω λίγο χρόνο. (comma, slight pause)

Your alternative word order is correct and natural.

How does the Greek present tense here correspond to English? Is it always I cook and I eat, or can it be I am cooking / I am eating?

Greek present covers both:

I cook (habitual)
I am cooking (right now)

Context decides. In your sentence:

μαγειρεύω στο σπίτι = I cook at home (habit, repeated)
τρώω έξω = I eat out (habit, repeated)

Because of όποτε (whenever), the habitual meaning is clear. So the best English is:
Whenever I have a bit of time, I cook at home instead of eating out.

Is there any difference between σπίτι and το σπίτι in sentences like this?

Yes:

το σπίτι = the house (with the article)
σπίτι (without article) often means home in a more general sense.

But when used with σε and contracted as στο σπίτι, it is the standard phrase for at home. In many contexts:

είμαι σπίτι = I am at home (more colloquial)
είμαι στο σπίτι = I am at home / in the house (neutral)

Here, μαγειρεύω στο σπίτι is the normal, idiomatic way to say I cook at home.