امروز در خانه هستم، اما فردا صبح زود به کار میروم.

در is the common preposition meaning in/at. So در خانه هستم is literally I am in/at home.
You can drop در in casual speech and writing and just say امروز خانه هستم, but امروز در خانه هستم sounds a bit more complete/neutral and is very common.

Yes. هستم is the 1st‑person singular form of بودن (to be): I am.
In Persian, you often keep هستم/هستی/است… (especially in more careful speech), whereas in very informal speech it can be shortened or implied depending on context.

Persian verb endings already show the subject, so هستم and می‌روم both clearly mean I.
Adding من is optional and usually used for emphasis or contrast, e.g. من امروز در خانه هستم (I (as for me) am home today).

اما is a common formal/neutral word for but/however. It links two contrasting clauses:

• clause 1: امروز در خانه هستم
• clause 2: اما فردا صبح زود به کار می‌روم

In casual speech, people also use ولی a lot for but. Both work here; اما can feel slightly more “written” than ولی.

It’s optional but very common. Persian punctuation often mirrors English: when you have two independent clauses joined by اما/ولی (but), a comma before it is natural:

• … هستم، اما … می‌روم.

You may also see it written without the comma, especially in informal texting.

Both are used for go to work, but:

• به کار می‌روم is straightforward and understandable everywhere.
• سرِ کار می‌روم (often written سر کار می‌روم) is extremely common in everyday Persian and can sound a bit more idiomatic, like “I’m going to work / heading to work.”

So your sentence could also be: … اما فردا صبح زود سرِ کار می‌روم.

Persian often uses the present tense with می- for planned or certain future events when a future time word is present (like فردا tomorrow).
So فردا … می‌روم is natural and can mean “I’m going / I will go tomorrow.”

There is a more explicitly future-like form using خواستن (خواهـ…), but in everyday speech the present is very common and often preferred.

In standard Persian orthography, می is a prefix and is ideally written with a half-space (zero-width non-joiner), like:

• می‌روم

In informal writing you’ll often see:

• میروم (no half-space)
  Both are readable, but می‌روم is the standard/formal spelling.

صبح زود means early in the morning. It’s an adverbial phrase.
Common placements include:

• فردا صبح زود به کار می‌روم (very natural)
• فردا به کار می‌روم، صبح زود (possible, but more like an afterthought)
• صبح زود فردا به کار می‌روم (also possible, slightly more emphasis on “early”)

Persian is generally SOV (subject–object–verb) and adverbs are flexible, but امروز هستم در خانه sounds unnatural.
The natural patterns are:

• امروز در خانه هستم
• امروز خانه هستم (more casual)

Putting هستم too early usually sounds awkward because Persian typically places the verb at the end of the clause.

A natural, connected pronunciation is roughly:

• fardâ sobh-e zud be kâr miram

Notes:

• فردا: fardâ
• Many speakers insert an -e- link after صبح in fast speech (sobh-e zud), even though it’s not always written.
• می‌روم in casual speech often sounds like miram (the vowel can reduce in speed), while careful speech is mi-ravam (more formal/pronounced).

It’s mostly neutral.

• Using اما can feel slightly more formal than ولی, but it’s still normal in conversation.
• Writing می‌روم with the half-space is more standard/formal orthography.
• If you want it more conversational, you could use ولی and سرِ کار:
  • امروز خونه‌ام، ولی فردا صبح زود سرِ کار می‌رم.