Κάθε πρωί χτενίζομαι πριν πάω στη δουλειά.

Κάθε πρωί means every morning.

Greek uses κάθε + singular noun to express every ...:

• κάθε μέρα = every day
• κάθε εβδομάδα = every week
• κάθε πρωί = every morning

So even though the idea is repeated, the noun stays singular after κάθε.

Χτενίζομαι is the mediopassive form, and here it has a reflexive meaning: I comb myself, which in natural English is usually I comb my hair.

This is very common in Greek with grooming and body-care verbs. Greek often uses the mediopassive where English uses myself or simply names the body part.

Compare:

• χτενίζομαι = I comb my hair / I comb myself
• χτενίζω τα μαλλιά μου = I comb my hair

Both are natural, but χτενίζομαι is a very common everyday way to say it.

Not here.

The ending -ομαι often looks like a passive form, but in Modern Greek these forms can also have a middle or reflexive meaning.

So in this sentence:

• χτενίζομαι does not mean I am being combed by someone
• it means I comb myself / I comb my hair

Context tells you which meaning is intended.

It is not stated explicitly, because Greek often leaves it understood when the meaning is obvious.

So:

• χτενίζομαι = literally something like I comb myself
• natural meaning: I comb my hair

If you want to be more explicit, you can say:

• Χτενίζω τα μαλλιά μου. = I comb my hair.

But in everyday Greek, χτενίζομαι is perfectly normal on its own.

After πριν meaning before, Greek normally uses a subjunctive-type verb form, not an ordinary present like πηγαίνω.

So:

• πριν πάω = before I go
• πριν πηγαίνω is not the normal choice here

The form πάω is used because the action is viewed as one complete event: the trip to work. Greek often does this after words like πριν, να, όταν in certain uses, etc.

Yes, they are closely related and both mean go.

In everyday Greek:

• πηγαίνω = go, be going
• πάω = go

But πάω is extremely common, and after forms like πριν it is the normal choice here.

A useful learner shortcut is:

• πηγαίνω often feels more ongoing / general
• πάω often feels more like a single going event

So πριν πάω στη δουλειά is the natural way to say before I go to work.

Because after πριν, Greek very often leaves να out.

So πριν πάω is completely normal and very common.

You may also encounter:

• πριν να πάω

but in everyday usage, πριν πάω is very natural.

Στη is the contraction of σε + τη(ν), so it means to the / at the, depending on context.

• σε = to / at / in
• στη = to the / at the

δουλειά means work or job.

So:

• πάω στη δουλειά = I go to work

With the verb πάω, it clearly means movement toward work, so English translates it as to work.

Because Greek usually drops subject pronouns when they are not needed.

The ending of χτενίζομαι already tells you the subject is I:

• -μαι = first person singular here

So εγώ is unnecessary unless you want emphasis or contrast:

• Εγώ χτενίζομαι, αλλά αυτός δεν χτενίζεται.
  I comb my hair, but he does not.

In a normal sentence, leaving out εγώ is the standard pattern.

No. Greek word order is more flexible than English word order.

The sentence

• Κάθε πρωί χτενίζομαι πριν πάω στη δουλειά.

is a natural, neutral way to say it:

• Κάθε πρωί = setting the time
• χτενίζομαι = main action
• πριν πάω στη δουλειά = before I go to work

You could also say:

• Χτενίζομαι κάθε πρωί πριν πάω στη δουλειά.

That is also natural. Greek often moves parts around for emphasis, rhythm, or focus, while the basic meaning stays the same.