Hoy hace frío, así que me pongo el gorro y la capucha antes de salir.

In Spanish, weather is often expressed with hacer:

• Hace frío = It’s cold
• Hace calor = It’s hot
• Hace sol = It’s sunny
• Hace viento = It’s windy

So hace frío is the normal way to talk about cold weather.
Es frío usually describes something as being cold by nature or characteristic, for example:

• El agua es fría = The water is cold
• Madrid es fría en invierno = Madrid is cold in winter

For general weather right now, hace frío is the natural choice.

Así que means so, therefore, or so then. It introduces a result:

• Hace frío, así que me pongo el gorro...
• It’s cold, so I put on my hat...

It is not the same as porque, which means because and gives the reason.

Compare:

• Hace frío, así que me pongo el gorro.
  It’s cold, so I put on my hat.

• Me pongo el gorro porque hace frío.
  I put on my hat because it’s cold.

Same basic idea, but the structure is different:

• así que = result
• porque = cause

Spanish often uses a reflexive pronoun with clothing:

• ponerse algo = to put something on

So:

• me pongo = I put on
• te pones = you put on
• se pone = he/she puts on

The me shows that the action is being done to oneself. In English we usually just say I put on my hat, but Spanish commonly says the equivalent of I put myself the hat.

Examples:

• Me pongo la chaqueta. = I put on my jacket.
• Se pone los zapatos. = He/She puts on shoes.

Without me, pongo just means I put or I place something somewhere:

• Pongo el libro en la mesa. = I put the book on the table.

With clothes, body parts, and personal items, Spanish often uses the definite article (el, la, los, las) instead of a possessive like my when it is already clear whose item it is.

So:

• me pongo el gorro literally looks like I put on the hat
• but it naturally means I put on my hat

The pronoun me already tells us the hat belongs to the speaker or is being worn by the speaker, so mi is often unnecessary.

This is very common:

• Me lavo las manos. = I wash my hands.
• Se quitó la chaqueta. = He/She took off his/her jacket.
• Nos ponemos los zapatos. = We put on our shoes.

You can say mi gorro if you want to emphasize that it is yours and not someone else’s.

They are different things:

• gorro = a hat, beanie, woolly hat, winter cap
• capucha = a hood

So in this sentence, the speaker is putting on both:

• a hat
• and the hood

That suggests they may be wearing a hooded coat or sweatshirt and are pulling up the hood as well.

In Spain, gorro often suggests a casual hat or warm hat rather than a formal hat.

Because Spanish normally repeats the article with each noun when listing things:

• el gorro y la capucha

That is the natural structure, just like:

• la camisa y el abrigo
• los guantes y la bufanda

You would not normally say el gorro y capucha here.

In Spanish, antes de is the standard structure before a noun or an infinitive.

• antes de salir = before going out / before leaving
• antes de comer = before eating
• antes de la cena = before dinner

So when a verb follows, Spanish uses:

• antes de + infinitive

That is why antes salir is incorrect.

After antes de, when the subject is the same as the main verb, Spanish uses the infinitive.

Here, the same person is doing both actions:

• me pongo = I put on
• salir = I go out

So Spanish says:

• antes de salir = before going out

This is very common:

• Voy a ducharme antes de dormir. = I’m going to shower before sleeping.
• Se lava las manos antes de comer. = He/She washes hands before eating.

If you want to mention a different subject, Spanish often uses a full clause:

• antes de que salgas = before you go out
• antes de que llegue Ana = before Ana arrives

The present tense here expresses a habitual or normal action:

• Hoy hace frío, así que me pongo...
• Today it’s cold, so I put on / I’m putting on...

In Spanish, the present tense can cover several ideas that English often separates:

• simple present: I put on
• present progressive in some contexts: I’m putting on
• habitual action: I usually put on

The exact meaning depends on context. Here it sounds very natural as either:

• what the speaker is doing now, or
• what they do in this situation

If Spanish wanted to emphasize the action in progress right now, it could also say:

• me estoy poniendo el gorro...

But the simple present is very normal and often preferred.

Sometimes, yes, but they are not always identical.

• así que strongly means so / therefore
• entonces can mean then, so, or in that case, depending on context

In this sentence:

• Hoy hace frío, así que me pongo el gorro... sounds very natural.
• Hoy hace frío, entonces me pongo el gorro... is understandable, but in many contexts así que sounds a bit more directly causal.

So así que is a very good choice here.

Both are possible depending on context.

Salir is a broad verb meaning:

• to leave
• to go out
• to go outside

In this sentence, because the speaker is putting on a hat and hood due to cold weather, before going out or before going outside is probably the best fit.

If the context were different, salir could also simply mean to leave:

• Salgo de casa a las ocho. = I leave home at eight.

Yes. In Spanish, ponerse la capucha is perfectly natural and means to put up the hood or to put on the hood.

Even though the hood is attached to the garment, Spanish still uses poner(se) for that action:

• Se puso la capucha porque llovía. = He/She put up the hood because it was raining.

So me pongo el gorro y la capucha sounds completely normal.

Yes, but the original order is very natural.

Original:

• Hoy hace frío, así que me pongo el gorro y la capucha antes de salir.

Possible variations:

• Como hoy hace frío, me pongo el gorro y la capucha antes de salir.
• Antes de salir, me pongo el gorro y la capucha porque hoy hace frío.

The original sentence flows well because it goes:

1. situation: Hoy hace frío
2. consequence: así que me pongo...
3. time reference: antes de salir

That makes it easy and idiomatic.