Després de descansar a casa, tornaré a respirar millor.

Després means afterwards/later, but when Catalan says after + noun or after + verb, it normally uses després de.

So:

• després = afterwards
• després de descansar = after resting

The de is part of the normal structure.

After després de, Catalan very often uses the infinitive when the subject is the same as in the main clause.

So this works like:

• Després de descansar... = after resting...

The person doing the resting is the same person who will breathe better later. This is very natural in Catalan.

Yes, but després de + infinitive is the neat, common way when the subject is the same.

A full clause is more likely if you need to state a different subject or make the sentence more explicit. In this sentence, the infinitive is the most natural choice.

A casa is a very common expression meaning at home or home. It usually does not take an article.

So:

• ser a casa = to be at home
• descansar a casa = to rest at home
• anar a casa = to go home

A la casa would usually mean to the house / at the house, referring to a specific building rather than the general idea of home.

By itself, tornar can mean to return. But here it is part of the very common pattern tornar a + infinitive, which means to do something again.

So:

• tornaré a respirar = I will breathe again

In this sentence, that idea combines with millor, so the sense is I will breathe better again.

Because the structure is fixed: tornar a + infinitive.

Examples:

• Torno a llegir-ho = I read it again
• Tornarà a ploure = It will rain again
• Tornaré a respirar = I will breathe again

That a is required in this construction.

It is the simple future, first person singular.

• tornaré = I will...
• tornaràs = you will...
• tornarà = he/she/it will...

So tornaré a respirar means I will breathe again.

Because millor is the normal word for better here.

With verbs like respirar, Catalan naturally uses millor:

• respirar millor = to breathe better

Més bé usually means something like rather or more accurately, so it would not sound natural in this sentence.

Catalan often leaves out subject pronouns because the verb ending already shows who the subject is.

• tornaré already tells you the subject is I

So jo is unnecessary unless you want extra emphasis or contrast:

• Jo tornaré a respirar millor, però ell no.

Yes, but the meaning changes slightly.

• respiraré millor = I will breathe better
• tornaré a respirar millor = I will breathe better again / I will get back to breathing better

So the version with tornar a + infinitive adds the idea of again or back to a previous state.

It separates the introductory time phrase from the main clause:

• Després de descansar a casa, = introductory phrase
• tornaré a respirar millor = main clause

This is similar to English punctuation. In short sentences, the comma is sometimes optional, but here it makes the sentence clearer and more natural.