Learn Hardcore Greek: Το αγαπημένο μου χόμπι είναι η τέχνη και μου αρέσει να ζωγραφίζω. - My favorite hobby is art and I like to draw/paint.

Putting μου in front (μου αγαπημένο χόμπι) is ungrammatical in Modern Greek.
If you want something before the noun, you use the stressed form δικός / δική / δικό:

το δικό μου αγαπημένο χόμπι (more emphatic: my favourite hobby, as opposed to someone else’s)

Why is αγαπημένο in the neuter form here?

Adjectives in Greek must agree in gender, number and case with the noun they describe.

χόμπι is grammatically neuter singular.
So the adjective αγαπημένος (favourite) must also be neuter singular, which is αγαπημένο.

Compare:

ο αγαπημένος φίλος (masculine: the favourite friend)
η αγαπημένη ταινία (feminine: the favourite movie)
το αγαπημένο χόμπι (neuter: the favourite hobby)

Because χόμπι is neuter, αγαπημένο must also be neuter.

What kind of word is χόμπι in Greek? What gender is it, and how do you make the plural?

χόμπι is a loanword from English hobby. In Modern Greek:

It is treated as a neuter noun.
It is indeclinable: its form does not change in different cases or in the plural.

So:

Singular: το χόμπι (the hobby)
Plural: τα χόμπι (the hobbies) — same form, plural shown only by the article τα and context.

In your sentence, Το αγαπημένο μου χόμπι is clearly singular because of:

το (singular article)
αγαπημένο (singular adjective)

Why do we use Το and η here? Could we drop the definite articles like in English My favourite hobby is art?

English often omits the article in places where Greek requires or prefers it.

With possessed nouns, Greek almost always uses the article:
- το βιβλίο μου (not just βιβλίο μου) = my book
- το αγαπημένο μου χόμπι = my favourite hobby
Dropping το here (Αγαπημένο μου χόμπι) is only possible in very special, often stylistic or exclamative contexts, not in a neutral sentence.
With abstract nouns used in a general sense, Greek usually uses the article:
- Η τέχνη είναι δύσκολη. = Art is hard.
- Η μουσική είναι η ζωή μου. = Music is my life.

So:

Το αγαπημένο μου χόμπι είναι η τέχνη
is the natural Greek equivalent of English My favourite hobby is art, even though English drops the articles.

Why is μου used twice, in αγαπημένο μου χόμπι and μου αρέσει? Is it doing the same job?

It’s the same little word μου, but it plays two different grammatical roles:

Possessive pronoun (my):
- το αγαπημένο μου χόμπι
  Here μου means my and shows possession of χόμπι.
Indirect object / experiencer with αρέσει:
- μου αρέσει
  Here μου means to me / I (as the person who experiences the liking).
  Literally: μου αρέσει ≈ it is pleasing to me.

So:

First μου = my (possessive)
Second μου = to me (person who likes something)

How does μου αρέσει work? Why don’t we say something like Εγώ αρέσω την τέχνη for I like art?

The verb αρέσω in Greek works almost the reverse way of English to like:

Structure: [something] αρέσει [σε κάποιον]
Literally: [something] is pleasing to [someone].

Examples:

Μου αρέσει η τέχνη.
Literally: Art is pleasing to me → I like art.
Μου αρέσεις.
Literally: You are pleasing to me → I like you (often romantically).

The subject of αρέσει / αρέσουν is the thing that is liked, not the person who likes it:

Μου αρέσει το χόμπι σου. = I like your hobby.
Μου αρέσουν οι ταινίες. = I like movies. (plural → αρέσουν)

So a sentence like Εγώ αρέσω την τέχνη is ungrammatical; αρέσω doesn’t take a direct object that way. You must use the μου αρέσει pattern.

Could we change the word order and say Η τέχνη είναι το αγαπημένο μου χόμπι? Is it the same?

Yes, you can say:

Το αγαπημένο μου χόμπι είναι η τέχνη.
Η τέχνη είναι το αγαπημένο μου χόμπι.

Both are grammatically correct and mean essentially the same: My favourite hobby is art.

The difference is mainly one of emphasis / focus:

Το αγαπημένο μου χόμπι είναι η τέχνη.
Starts from my favourite hobby and then identifies it as art.
Η τέχνη είναι το αγαπημένο μου χόμπι.
Starts from art and says art is my favourite hobby (slightly more emphasis on art).

Greek word order is relatively flexible, and both versions sound natural.

Why do we need να before ζωγραφίζω? Why not just μου αρέσει ζωγραφίζω?

Modern Greek does not have an infinitive form like English to paint or painting in this sense. Instead, it uses:

να + finite verb (subjunctive) to express to do something after verbs like like, want, can, start, try, etc.

So:

μου αρέσει να ζωγραφίζω ≈ I like to paint / I like painting
θέλω να ζωγραφίζω = I want to be painting / to paint (habitually)
μπορώ να ζωγραφίσω = I can paint (once / successfully)

Without να, ζωγραφίζω is just a normal present tense verb form meaning I paint.

You cannot say μου αρέσει ζωγραφίζω; it is ungrammatical.
The correct pattern is always μου αρέσει να + verb.

What is the difference between να ζωγραφίζω and να ζωγραφίσω?

Both are subjunctive forms of ζωγραφίζω (to paint), but they express different aspect:

να ζωγραφίζω – present subjunctive
Focus on ongoing / repeated / habitual action.
With μου αρέσει, it means you like the activity in general:
- Μου αρέσει να ζωγραφίζω.
  I like painting (as a hobby, in general).
να ζωγραφίσω – aorist subjunctive
Focus on a single, complete event. Often used with verbs like θέλω (I want), πρέπει (I must), etc.:
- Θέλω να ζωγραφίσω έναν πίνακα.
  I want to paint a (single) picture.

With μου αρέσει, you almost always use the present form (να ζωγραφίζω) for hobbies and general likes.

What is the difference between τέχνη, ζωγραφική, and ζωγραφίζω?

They are related but not the same:

η τέχνη = art in general, or sometimes a particular art form depending on context.
In your sentence, it’s general: art as a broad field.
η ζωγραφική = painting (as an art form/discipline).
You could say:
- Το αγαπημένο μου χόμπι είναι η ζωγραφική.
  My favourite hobby is painting.
ζωγραφίζω = I paint / I draw (the verb).
In μου αρέσει να ζωγραφίζω, it refers to the activity of painting/drawing.

So your sentence combines a general statement:

η τέχνη (art) as the favourite hobby

with a specific activity:

να ζωγραφίζω (to paint/draw) as something you like doing.