Learn Hardcore Greek: Τώρα βρίσκομαι στο σπίτι και χαλαρώνω.

Greek usually leaves out subject pronouns like εγώ (I) because the verb ending already shows the person and number.
In βρίσκομαι and χαλαρώνω, the -μαι and -ω endings tell you it is 1st person singular (I).
So Τώρα βρίσκομαι στο σπίτι και χαλαρώνω is naturally understood as Now I am at home and (I) am relaxing, without needing εγώ.

What does βρίσκομαι mean, and how is it different from είμαι?

Βρίσκομαι literally means I am located / I am found (somewhere).
Είμαι is the general verb to be.
Very often they can both be used for location, but:

Είμαι στο σπίτι = I am at home.
Βρίσκομαι στο σπίτι = I am (currently / physically) located at home, sometimes sounding a bit more formal or descriptive.
You will hear είμαι more in everyday speech, but βρίσκομαι is also common and correct.

Can I just say Τώρα είμαι στο σπίτι και χαλαρώνω instead? Does it sound natural?

Yes, Τώρα είμαι στο σπίτι και χαλαρώνω is completely natural and very common.
In casual conversation, many speakers would actually prefer είμαι here.
Using βρίσκομαι just adds a slight nuance of “I am (situated) at home right now,” but the meaning is essentially the same.

Why does βρίσκομαι end in -ομαι? Is it reflexive or passive?

The ending -ομαι is the middle/passive ending in the present tense.
Βρίσκω = I find (active),
βρίσκομαι = I am found / I am located (middle/passive form, but with its own meaning).
In this sentence it is not reflexive like “I find myself”; it is just the normal way to say I am located somewhere.

What exactly does χαλαρώνω mean?

Χαλαρώνω means I relax / I am relaxing / I unwind.
It can be used:

For people: Χαλαρώνω στο σπίτι. = I relax at home.
For things: Το σχοινί χαλαρώνει. = The rope is loosening.
Here it clearly means I am relaxing / unwinding.

The English uses “I am relaxing.” Why is Greek just using the simple present χαλαρώνω?

Modern Greek has one present tense that covers both:

I relax (habitual)
I am relaxing (right now)
So χαλαρώνω can mean either, depending on context.
With Τώρα (now) in the sentence, it clearly means I am relaxing (right now).

What does στο mean, and where does it come from?

Στο is a contraction of σε + το:

σε = in, at, to
το = the (neuter singular)
So στο σπίτι literally is in the house.
Similar contractions:
σε + τον → στον
σε + την → στη(ν)

Why is it στο σπίτι? Could it also be just σπίτι without στο?

Both exist, but they are used slightly differently:

Είμαι στο σπίτι. = I am in the house / at home (more literal, includes στο and the article).
Είμαι σπίτι. = I am at home (very idiomatic, no preposition or article).

In your sentence, στο σπίτι is perfectly normal and perhaps a bit more neutral.
Saying Τώρα είμαι σπίτι και χαλαρώνω is also very common and sounds very natural.

Can I change the word order, for example to Τώρα χαλαρώνω στο σπίτι?

Yes. Greek word order is flexible, as long as the grammar (endings, prepositions, articles) is correct.
All of these are fine and natural, with slightly different emphasis:

Τώρα βρίσκομαι στο σπίτι και χαλαρώνω.
Τώρα είμαι στο σπίτι και χαλαρώνω.
Τώρα χαλαρώνω στο σπίτι. (focus more on the relaxing)
Χαλαρώνω τώρα στο σπίτι. (similar, with a bit more emphasis on τώρα after the verb)

What does και do here? Could it also mean “also”?

In this sentence, και is a simple and, joining two verbs with the same subject:

βρίσκομαι … και χαλαρώνω = I am (located) … and (I) am relaxing.

Και can also mean also / even, but then its position changes, e.g.

Και χαλαρώνω. = I also relax.
Here, because it is between the two verbs, it clearly means and.

How do you pronounce Τώρα βρίσκομαι στο σπίτι και χαλαρώνω?

A simple pronunciation guide (stressed syllables in capitals):

Τώρα → TO-ra
βρίσκομαι → VRIS-ko-me
στο → sto
σπίτι → SPEE-ti
και → ke (a bit like “keh”, often slightly palatal, between k and ky)
χαλαρώνω → ha-la-RO-no

Altogether: TO-ra VRIS-ko-me sto SPEE-ti ke ha-la-RO-no.