Estoy un poco enojado hoy.

Spanish uses estar (estoy) for temporary states, emotions, and conditions. Being angry today is a temporary feeling, so you say estoy enojado.
Ser (soy) is for more permanent traits (nationality, profession, character). Saying soy enojado would sound like “I am an angry person (by nature),” which is not what this sentence means.

You normally just say Estoy un poco enojado hoy.
In Spanish, the subject pronoun (yo) is usually dropped because the verb ending (-oy in estoy) already shows who the subject is.
You’d only add yo for emphasis or contrast, for example:

• Yo estoy un poco enojado hoy, pero ella está muy tranquila.
  “I am a bit angry today, but she is very calm.”

Yes, enojado/enojada agrees with the gender of the person who is angry:

• A man: Estoy un poco enojado hoy.
• A woman: Estoy un poco enojada hoy.

If the speaker is nonbinary, usage varies by country and community, but you may see forms like enojade in some inclusive-language contexts. In standard grammar, it matches masculine or feminine.

Enojado usually means angry, but the strength depends on context:

• Estoy enojado alone can sound stronger: “I’m angry.”
• Estoy un poco enojado softens it: “I’m a bit/ kind of angry.”

Other options with different nuances:

• Estoy molesto. – I’m upset / annoyed. (Often a bit softer.)
• Estoy de mal humor. – I’m in a bad mood.

In Latin America:

• Enojado – very common for “angry, mad” in many countries (Mexico, much of Central and South America).
• Enfadado – understood but less common in many Latin American countries; more frequent in Spain. In some areas it can sound more formal or old-fashioned.
• Molesto – often “annoyed, bothered, upset.” Sometimes softer than enojado, but in some contexts it can still be strong.

So Estoy un poco enojado hoy sounds perfectly natural in most of Latin America.

Yes, both are correct:

• Estoy un poco enojado hoy.
• Hoy estoy un poco enojado.

Putting hoy at the beginning (Hoy estoy…) is very common and slightly emphasizes “today”. The meaning is the same; it’s mainly a matter of rhythm and what you want to highlight.

They look similar but feel different:

• un poco enojado ≈ “a little / a bit angry.” (Neutral.)
• poco enojado is rare here and sounds odd; by itself poco often means “not very” or “hardly.”

So you use un poco to soften the adjective and sound more natural:

• Estoy un poco enojado. – I’m a bit angry.

Yes:

• Estoy un poquito enojado hoy.

Un poquito is the diminutive of un poco. It usually sounds even softer or more informal, like “just a tiny bit” or “a little bit.” It can also sound a bit cuter or more playful depending on tone.

• Estoy un poco enojado hoy. → “I’m a bit angry today.” (softer, more moderate)
• Estoy enojado hoy. → “I’m angry today.” (stronger, more direct)

Adding un poco makes your emotion sound less intense and more polite or understated.

It’s grammatically possible, but it sounds less natural. The usual options are:

• Hoy estoy un poco enojado.
• Estoy un poco enojado hoy.

Spanish prefers time expressions like hoy either at the beginning or the end of the sentence, not usually in the middle between estoy and the rest.

In this sentence, hoy is just telling you when you are angry (today), not the cause or purpose.

• enojado hoy = “angry today.”
  The prepositions por and para would change the meaning:
• enojado por algo – angry about/because of something.
• enojado para hoy – doesn’t make sense here; para is for purpose, deadlines, recipients, etc.

So you simply add hoy as a time expression without a preposition.