Learn Hardcore Russian: Они то обсуждают будущий отпуск, то спорят о деньгах, но потом всё‑таки находят компромисс. - They either discuss the future vacation or argue about money, but then still find a compromise.

The correlative particles то … то … express alternation: one thing happens, then another, often repeatedly or in a back‑and‑forth way.
In this sentence it’s like saying:

“Now they’re discussing their future vacation, now they’re arguing about money”, or
“They keep either discussing the upcoming vacation or arguing about money”.

It suggests switching between these actions, not just a single sequence once.

Is this то the same word as the pronoun то meaning “that”?

Formally it’s the same word, but here it functions as a particle in a fixed pair то … то …, not as a pronoun (“that”).
You don’t translate it as “that” in English; you translate the whole construction as “now … now …”, “sometimes … sometimes …”, or “either … or …” depending on context.

Why are all the verbs in the present tense? Does this describe something habitual?

Yes. Russian present tense is often used for:

current actions (what’s happening right now), and
repeated / typical actions (what they tend to do).

Here, with то … то … and потом всё‑таки находят, it naturally reads as a habitual pattern:

“They (tend to) discuss their future vacation, then argue about money, but in the end they still find a compromise.”

In English we’d usually use the present simple to show this kind of typical behavior, which matches the Russian present here.

Why is it обсуждают будущий отпуск, not something like обсуждают о будущем отпуске?

The verb обсуждать (“to discuss”) takes a direct object in the accusative case, without a preposition:

обсуждать что? — будущий отпуск (Accusative masculine singular)

Using о would be ungrammatical with обсуждать.
If you wanted to use о, you’d need a different verb:

говорить о будущем отпуске – “to talk about the future vacation”
спорить о будущем отпуске – “to argue about the future vacation”

What’s the nuance of будущий отпуск? Could you also say следующий отпуск?

Both are correct, but slightly different:

будущий отпуск – literally “future vacation”; neutral, just “the vacation that is ahead of them”.
следующий отпуск – “next vacation”; focuses more on sequence (this one, then the next one, etc.).

In many everyday contexts they can be interchangeable, but будущий отпуск is a very standard way to say “upcoming vacation” when you’re talking about the one that’s already planned or expected.

What’s the difference between отпуск and каникулы?

отпуск is mainly vacation from work (for adults with a job).
- У меня отпуск в июле. – I have vacation in July.
каникулы is mainly school / university holidays or regular breaks.
- Летние каникулы – summer holidays (for students).

So будущий отпуск here implies their vacation from work, not school holidays.

Why is it спорят о деньгах and not спорят деньги or something similar?

The verb спорить (“to argue”) usually takes о + Prepositional case to indicate the topic:

спорить о чём? — о деньгах (about money)

The noun деньги (“money”) is plural; its Prepositional plural is деньгах:

Nom. Pl.: деньги
Prep. Pl.: о деньгах

So о деньги would be wrong; the correct form is о деньгах.

What does всё‑таки mean here, and what nuance does it add?

всё‑таки means roughly “nevertheless / all the same / in the end after all”.
Here it signals that despite all the discussions and arguments, the result is still positive:

“…but then they still manage to find a compromise (in the end, after all that).”

Without всё‑таки, the sentence would just state the fact that they find a compromise; всё‑таки emphasizes that this is a bit surprising or noteworthy given the preceding conflict.
A close synonym is всё же.

Why is всё‑таки written with a hyphen? Can it be written separately?

The standard modern spelling is всё‑таки with a hyphen, because it functions as a single adverbial particle.
In older texts you may see всё таки as two words, but in contemporary standard Russian the hyphenated form is considered correct.
So you should learn and use всё‑таки.

Why is it находят компромисс (imperfective) and not найдут компромисс (perfective future)?

находят is imperfective present (“they find / they keep finding”), which fits a repeated or typical outcome.
найдут would be perfective future (“they will find (once, at some point)”).

The whole sentence describes a recurring pattern of behavior, so Russian uses the imperfective present: this is what typically happens when they interact.
If you were talking about one specific situation in the future, найдут компромисс would be more natural.

Why is there a comma after будущий отпуск before то спорят о деньгах?

In то обсуждают …, то спорят …, you have two parallel (homogeneous) predicates tied by the repeated particle то.
In Russian, when a conjunction or particle is repeated between homogeneous parts, you normally put a comma between those parts:

то …, то …
или …, или …
не то …, не то …

So the comma after будущий отпуск separates то обсуждают будущий отпуск from то спорят о деньгах.

Why isn’t они repeated before находят компромисс?

Russian doesn’t need to repeat the subject if it remains the same and is clear from context.
Here они is the subject for all three predicates:

(они) то обсуждают…
(они) то спорят…
(они) находят компромисс.

Repeating они before находят is possible but unnecessary and would sound heavier; the usual style is to state the subject once and then omit it when it’s obvious.

Where is the stress in the less obvious words of this sentence?

Stresses (marked with ´ over the stressed vowel):

они́
обсужда́ют
бу́дущий
о́тпуск
спо́рят
о деньга́х
пото́м
всё‑та́ки
находя́т (colloquial) / нахóдят (also heard; dictionaries usually give нахо́дят)
компроми́сс

In careful dictionary pronunciation, нахо́дят is standard; in fast speech many people shift stress slightly, but you should aim for нахо́дят компроми́сс.