Learn Hardcore Japanese: kazi to sigoto wo suruno ha mendou da ga, kuhuusureba daitai dekiru. - Doing both housework and work is troublesome, but with some ingenuity it’s mostly doable.

の nominalizes the verb phrase, turning 家事と仕事をする into “the act of doing housework and work,” so it can take は and be the topic: …するのは….
You could also say …することは…. Nuance:

の: a bit more colloquial and concrete (the actual doing).
こと: a bit more formal/abstract (the concept of doing).
Both are fine here: 家事と仕事をするのは面倒だ… ≈ 家事と仕事をすることは面倒だ….

Why は in するのは instead of が? What’s the difference between …するのは… and …するのが…?

…するのは… marks the whole clause as the topic, setting up a contrast/comment that follows. It fits naturally with the contrastive だが later.
…するのが… makes it the grammatical subject of the predicate; it’s more “descriptive” and not as contrastive.
You could say …するのが面倒だが、…, but は better highlights “as for doing X, it’s a hassle, but…”.

Why と in 家事と仕事 and not や? What’s the nuance?

AとB lists items exhaustively: “A and B (specifically those).”
AやB is non-exhaustive: “A and B (and things like that).”
Here と suggests “housework and (paid) work” as the set in question. 家事や仕事 would broaden it to “housework, work, etc.”

Do I need to repeat を for both nouns? Is 家事を仕事をする ever correct?

No. When listing multiple direct objects with と or や, attach the particle once at the end: 家事と仕事をする / 家事や仕事をする.
家事を仕事をする is ungrammatical in standard usage.

What exactly does 面倒だ mean? How is it different from 面倒くさい or 大変だ?

面倒だ: “a hassle,” “troublesome” (mild complaint, neutral register).
面倒くさい: stronger, more emotive “ugh, such a pain.”
大変だ: “hard/difficult,” emphasizes effort or severity, not just annoyance.
Note: 面倒を見る means “to take care of (someone),” unrelated to “a hassle.”

What’s the tone of だが compared with が, けど, or ですが?

だが: written-ish, somewhat stiff/plain style.
が (as a connector): neutral; pairs well with both だ/です depending on the sentence.
けど/けれど(も): more conversational.
ですが: polite style.
Keep style consistent:
Plain: 面倒だが、…できる。
Polite: 面倒ですが、…できます。

What does 工夫する really mean here?

工夫する is “to devise/work out a clever way,” “to be resourceful,” “to make tweaks/adjustments.” It implies finding practical, creative ways to manage. Examples:

時間の使い方を工夫する: find smarter ways to use time.
安く済むように工夫する: figure out cost-saving tweaks.

Why use the ば conditional (工夫すれば) instead of したら, すると, or なら?

…すれば: general/hypothetical condition; good for advice or general truths.
…したら: more event/sequence-oriented (“when/after you do…”).
…すると: habitual/immediate result (“when you do X, Y happens”).
…なら: “if it’s the case that…” (given condition).
Here 工夫すれば nicely expresses “if you apply some ingenuity (in general), then…”

Who is the subject of できる? It isn’t stated—what does it refer to?

It’s omitted. The topic is established by 家事と仕事をするのは. So できる means “(one/you/I) can mostly do it.” Japanese often drops subjects when they’re clear from context.

I learned できる usually takes が (e.g., 日本語ができる). Why do we have を earlier in the sentence?

The を belongs inside the nominalized clause 家事と仕事をする (do housework and work). The later できる stands on its own and refers back to that topic. You’re not saying をできる (which is generally wrong); you’re saying “As for doing A and B, (one) can do it.”
A compact rephrase is: 家事や仕事は、工夫すればだいたいできる。

What nuance does だいたい have? How does it differ from ほとんど, たいてい, or ほぼ? Where can it go?

だいたい: “mostly/roughly/approximately.” Flexible and casual.
ほとんど/ほぼ: “almost/nearly (all),” closer to ~90–99%.
たいてい: “usually/for the most part” (habitual frequency).
Placement: 工夫すればだいたいできる is natural. だいたいはできる adds a contrastive nuance (“at least mostly, one can”).

Could I say …することができる instead of できる?

Grammatically yes, but it’s heavier. In everyday speech/writing, prefer できる (or the potential form of the main verb). Also, repeating する here would be clunky after …するのは…. A tidy alternative is: 家事や仕事は、工夫すればだいたいできる。

Are the spaces in the sentence normal in Japanese?

No. Standard Japanese doesn’t use spaces between words. They were added for learners. Normally you’d write: 家事と仕事をするのは面倒だが、工夫すればだいたいできる。

Could I explicitly say それ in the second clause?

You can for emphasis, e.g., …が、工夫すればそれはだいたいできる。 Usually it’s omitted because the topic is clear; leaving it out sounds more natural: …が、工夫すればだいたいできる。

Can I drop だ in 面倒だ?

In normal sentences, no—you need the copula: 面倒だ (plain) or 面倒です (polite). Dropping it is limited to headlines, notes, or very casual fragments.