Learn Hardcore Chinese: tā xiànzài zài jiā zuòfàn ne.

It marks that the action/situation is ongoing right now and softens the tone. Here it’s like saying “(she’s) cooking (right now).” It is not a question particle in this sentence.

Is 在 here the progressive marker or just “at/in”?

Here 在 is part of 在家 (“at home”), so it marks location. The “in-progress” feel comes from 呢 (and context). If you want the progressive marker 在, put it before the verb: 她现在在做饭呢. With a location, people usually avoid a double 在 and say 她现在在家做饭呢 or 她现在在家正在做饭.

Can I drop 呢? What changes?

Yes. 她现在在家做饭 (without 呢) can still be read as a current fact, but it is also compatible with a “nowadays/these days” meaning. Adding 呢 (or using 在/正在) nudges it toward “right now, in progress.”

Do I need both progressive 在/正在 and 呢?

No. Any of these works:

在 + V: 她现在在做饭。
正在 + V: 她现在正在做饭。
V + 呢: 她现在做饭呢。 They’re often combined for a colloquial, emphatic feel: 她现在在做饭呢 / 她现在正在做饭呢. With a place, avoid clunky 她现在在家在做饭呢; prefer 她现在在家做饭呢 or 她现在在家正在做饭(呢).

Why isn’t there a 是 (“to be”) like in English “is cooking”?

Chinese doesn’t use 是 to form the progressive. 是 links nouns (她是老师). Progressive meaning is expressed with 在/正在/呢. A contrastive, emphatic form like 她现在是在家做饭呢 is possible but marked.

Is the word order typical? Where do time and place go?

Yes. Typical order is: Subject + Time + Place + Verb(+Object) + Particle. So: 她 + 现在 + 在家 + 做饭 + 呢. You can front 现在 for emphasis: 现在她在家做饭呢.

Where should 正在 go if I also mention the place?

Put it right before the action part. Natural options:

她现在正在做饭。
她现在正在家里做饭。
她现在在家里正在做饭。 (acceptable but a bit heavier) The first two are most common.

How is 在家 different from 在家里?

Both mean “at home.” 在家 is slightly shorter/more colloquial; 在家里 is a bit more explicit. No real difference in meaning here.

What’s the nuance of 做饭, 做菜, 煮饭, and 烧饭?

做饭: general “cook/make a meal” (neutral, common in mainland).
做菜: “cook dishes,” focuses on preparing dishes.
煮饭: literally “cook rice” (also “cook” in Taiwan usage).
烧饭: “cook (a meal)” in some southern regions/dialects.

Does 在家 mean “at her own home,” or could it be someone else’s?

By default with subject 她, 在家 is understood as “at her (own) home.” If it’s someone else’s, it’s usually specified: 在她朋友家, 在老师家, etc.

How do I negate this meaning?

Not doing it now (progressive negation): 她现在没在做饭 or 她现在没在家做饭. Use 没(有) with the progressive.
Not at home now (location negation): 她现在不在家。
Habitual/changed situation: 她现在不做饭了 (“she doesn’t cook now/anymore”). Avoid 不在 + V for progressive negation; it’s usually read as “not at (a place)” instead.

How do I make it a yes/no question? Can I keep 呢?

Use 吗 and drop 呢: 她现在在家做饭吗？ Don’t use 吗 and 呢 together at the end.

How do I ask “What is she doing now?”

Use 什么: 她现在在做什么(呢)？ If you want to keep the place: 她现在在家做什么(呢)？

How do I say “Now she cooks at home” as a new situation, or “She started cooking (now)”?

Use 了 for change of state:

她现在在家做饭了。 (Now she cooks at home — new situation.)
她现在开始做饭了。 (She has started cooking now.)

Any pronunciation tips for the line?

Pinyin: tā xiànzài zài jiā zuòfàn ne. 呢 is neutral tone. Keep both syllables of 做饭 in 4th tone (zuò-fàn).