Moj brat se obrije prije posla, ali se ja samo umijem i operem zube.

Se is a reflexive clitic. It shows that the action is done to the subject.

• obrijati se = to shave oneself / to shave
• umiti se = to wash one’s face

In English, we often do not translate se separately, but in Croatian it is an important part of these verbs.

Because se is a clitic, and Croatian clitics usually go in the second position of a clause.

• Moj brat se obrije...
  The first unit is Moj brat, so se comes after it.
• ali se ja samo umijem...
  The clause begins with ali, so se comes right after ali.

This is normal Croatian word order for clitics.

These forms are from perfective verbs:

• obrijati se = shave completely
• umiti se = wash one’s face
• oprati = wash completely

Perfective verbs present the action as a completed whole. In a sentence about routine actions, this can make the actions sound like completed steps in a sequence.

If you used the imperfective forms:

• brije se
• umivam se
• perem zube

the focus would be more on the ongoing/habitual activity itself.

No. Umijem se specifically means I wash my face.

For general washing, Croatian would use other verbs, for example:

• perem se = I wash myself
• tuširam se = I shower

So umiti se is more specific than English wash.

Because prije takes the genitive case.

The noun posao changes like this:

• nominative: posao
• genitive: posla

So:

• prije posla = before work

This is a regular pattern after prije.

Because oprati zube is not reflexive. Here zube is the direct object of the verb:

• operem zube = I wash/brush my teeth

The se belongs only to umijem, because umiti se is reflexive. It does not belong to operem.

Croatian often leaves out possessives with body parts when the owner is obvious.

So operem zube naturally means I brush/wash my teeth.

You can say operem svoje zube or operem moje zube, but that is usually unnecessary unless you want extra emphasis or contrast.

Yes, literally it is closer to wash my teeth, but in normal Croatian this is the standard way to express brush my teeth.

So:

• prati / oprati zube = to brush one’s teeth

This is one of those places where Croatian and English use different everyday wording.

Yes, it could be omitted, because the verb form already shows first person singular.

But ja is included for contrast:

• Moj brat... ali se ja...
  = My brother..., but I...

So the pronoun is there to emphasize the difference between the brother’s routine and the speaker’s routine.

Sometimes yes, but not freely.

For example, Croatian can move words around for emphasis, but clitics like se still have to stay in their usual clitic position. That is why:

• Moj brat se obrije prije posla sounds normal.
• Ali se ja samo umijem i operem zube also sounds normal.

A version like Ali ja se samo umijem... is much less natural in standard Croatian. So word order is somewhat flexible, but clitic placement strongly limits it.