Learn Hardcore Esperanto: En la venonta semajno mi ne laboros en la oficejo. - In the coming week I will not work in the office.

They mean the same thing here.
en la venonta semajno sounds a bit more explicit and slightly more formal or careful; venontsemajne is shorter and very common in everyday Esperanto.

What exactly is venonta? Is it related to the verb veni (to come)?

Yes. Venonta is the future active participle of veni (to come).

veni = to come
venanta = coming (right now / ongoing)
veninta = having come / who has come
venonta = going to come / that will come

In en la venonta semajno, it literally means in the week that is going to come → in the coming week → next week.

So venonta behaves like an adjective and must agree in number and case if needed:

venonta semajno = the coming week
venontaj semajnoj = the coming weeks

Why is there la before venonta semajno and oficejo? In English we don’t say the next week or the office here.

Esperanto uses the definite article la more consistently than English.

en la venonta semajno = in the coming week
Here we are speaking about a specific, known week (the one directly after this week), so la is natural.
en la oficejo = in the office
Again, this is usually a specific office that speaker and listener know about (e.g. my usual workplace), so la fits.

In Esperanto, you typically do use la for:

unique or context‑specific things: la suno, la lernejo, la oficejo (my/our usual office)
time expressions with a specific period: en la venonta semajno, dum la somero

Leaving la out (e.g. en venonta semajno) is possible but sounds less natural here, and some speakers would even find it a bit odd.

Is the word order fixed? Could I say Mi ne laboros en la oficejo en la venonta semajno instead?

The word order in Esperanto is quite flexible. All of these are grammatically correct and mean the same:

En la venonta semajno mi ne laboros en la oficejo.
Mi ne laboros en la oficejo en la venonta semajno.
Mi en la venonta semajno ne laboros en la oficejo.
En la venonta semajno en la oficejo mi ne laboros. (more unusual, but still correct)

The basic guideline:

Keep ne directly before the verb it negates (ne laboros).
Move time and place phrases (en la venonta semajno, en la oficejo) around for emphasis or style.

The original version puts the time at the beginning, which is very common and clear.

Why is ne before laboros? Could I put ne somewhere else?

In Esperanto, ne normally appears immediately before the verb (or the specific word/phrase it negates).

mi ne laboros = I will not work
mi laboros = I will work

If you move ne far from laboros, it sounds wrong or can change the focus of the negation. These are not natural:

✗ mi laboros ne en la oficejo (understandable, but feels clumsy)
✗ ne mi laboros en la oficejo (now it means it won’t be me who will work in the office)

So: for a simple sentence like this, keep ne directly before the verb: mi ne laboros.

How does the future tense ending -os work? Why not say something like estos laboranta?

Esperanto has very regular verb endings:

-as = present (I work / am working) → mi laboras
-is = past (I worked / was working) → mi laboris
-os = future (I will work / will be working) → mi laboros

For normal future actions, you just use -os. You do not need esti plus a participle to make the future.

mi ne laboros en la oficejo = I will not work in the office (or: I won’t be working in the office)

Forms like mi estos laboranta are possible but more complicated and used only when you really want to emphasize an ongoing process at a particular future time. In everyday speech, mi ne laboros is exactly what you need.

Could I say Mi ne laboros oficeje instead of en la oficejo? What’s the difference?

Yes, you can, but there is a nuance.

en la oficejo = physically in the office (a concrete place)
oficeje = in an office‑like way / at an office‑type workplace, more adverbial and a bit more abstract

In practice:

Mi ne laboros en la oficejo – I won’t be at that specific office working there.
Mi ne laboros oficeje – I won’t be working in an office setting (maybe I’ll work from home, or not do office‑type work).

Most of the time when you mean your usual workplace building, en la oficejo is the most straightforward and common choice.

Can I drop mi like you can drop pronouns in Spanish (e.g. just say Ne laboros en la oficejo)?

No. In standard Esperanto you normally do not drop the subject pronoun.

Mi ne laboros en la oficejo. – correct and normal
Ne laboros en la oficejo. – sounds incomplete/odd; we don’t know who won’t work

Because verbs do not change according to person (there is no difference between I work, he works, etc.), you need the pronoun to show who the subject is:

mi laboros – I will work
li laboros – he will work
ili laboros – they will work

So always include mi, vi, li, ŝi, ĝi, ni, ili, or a noun as the subject.

What is the difference between en la venonta semajno and dum la venonta semajno?

Both can sometimes be translated as next week, but they focus on slightly different things:

en la venonta semajno = in the coming week, placing the action within that week.
dum la venonta semajno = during the coming week, emphasizing the duration / time span.

In your sentence:

En la venonta semajno mi ne laboros en la oficejo.
→ At that time (next week), I will not work in the office.
Dum la venonta semajno mi ne laboros en la oficejo.
→ Throughout that time span (for the duration of next week), I will not work in the office.

In many contexts they’ll be understood the same, but dum makes you think a bit more about the whole period.

If this is a plan or decision (not just a prediction), do I still use laboros?

Yes. Esperanto -os covers all the main future uses that English splits into will and going to:

future fact / prediction
already‑made plan
intention or decision

So:

Mi ne laboros en la oficejo.
can mean:
- I will not work in the office. (prediction)
- I’m not going to work in the office. (plan/intention)

Context and additional words show your attitude if needed:

Mi decidis, ke mi ne laboros en la oficejo. – I have decided I will not work in the office.
Mi planas ne labori en la oficejo en la venonta semajno. – I plan not to work in the office next week.