Learn Hardcore Chinese: tā duì kēxué hé shùxué dōu hěn yǒu xìngqù. - She is very interested in science and mathematics.

With 都: highlights that each of the mentioned subjects fits the description.
Without 都: still means she likes these two things, but the “both/all” flavor is weaker.

Native speakers often include 都 in this kind of sentence because it sounds natural and clear when talking about multiple items.

Why do we have both 很 (hěn) and 有 (yǒu)? Is 很 really “very” here, and is it necessary?

The core structure is:

有兴趣 – to have interest
很有兴趣 – to have a lot of interest / to be very interested

有 is a verb meaning to have.
兴趣 is a noun meaning interest.
很 is a degree adverb meaning very, but in Chinese it is often weaker than English “very”.

Here:

很有兴趣 can be understood as “is (quite) interested”, more than a neutral “has interest”.

Is 很 necessary?

Grammatically, you can say 她对科学和数学有兴趣, and it is possible.
But native speakers very often add 很 (or another degree word) before this kind of phrase; 有兴趣 / 高兴 / 喜欢 etc. sound more natural with a degree word in many everyday sentences.

So 很 here both:

Softens and naturalizes the adjective–like phrase, and
Adds a degree of emphasis, roughly like “quite / really / very interested”.

What is the difference between 有兴趣 and 感兴趣?

Both can translate as “to be interested in”, and in many contexts they are interchangeable.

有兴趣 (yǒu xìngqù)
- Literally: to have interest
- Slightly more like stating a general interest or inclination.
- Common in talking about whether you want to do / are open to something.
- Example:
  - 你对学中文有兴趣吗？ – Are you interested in learning Chinese?
感兴趣 (gǎn xìngqù)
- Literally: to feel interest
- Emphasizes the feeling of interest, often a bit more subjective.
- Also very common and natural.
- Example:
  - 她对科学很感兴趣。 – She is (really) interested in science.

In your sentence, both are fine:

她对科学和数学都很有兴趣。
她对科学和数学都很感兴趣。

The difference is subtle; 感兴趣 might sound a bit more like describing her emotional interest, while 有兴趣 can feel a bit more like “has an interest / inclination”.

Can I use 跟 (gēn) instead of 和 (hé) between 科学 and 数学?

Yes. In most everyday spoken Mandarin, 和 and 跟 are more or less interchangeable when they mean “and” between nouns:

她对科学跟数学都很有兴趣。
她对科学和数学都很有兴趣。

Both are natural.

Very rough tendencies:

和 – slightly more neutral, common in both spoken and written language.
跟 – very common in spoken language, may feel a bit more informal.

There is also 及 (jí), which is more formal/written, often in documents or more serious writing:

她对科学及数学都很有兴趣。 – Quite formal style.

Why is it 都 here and not 也 (yě)? What’s the difference?

都 and 也 both translate to things like “all / both / also / too”, but they are used differently.

都 (dōu) – used when you have a set of two or more items, and you want to say the predicate applies to all of them.
- Pattern: (Subject) + (multiple items) + 都 + predicate
- Example: 我哥哥和妹妹都喜欢音乐。 – My older brother and younger sister both like music.
也 (yě) – means also / too, usually adding a new item to something mentioned before.
- Pattern: Subject + 也 + predicate
- Example:
  - 他喜欢科学。她也喜欢科学。
    – He likes science. She also likes science.

In your sentence:

You are talking about two subjects (science and math) together in one clause, so 都 is appropriate:
她对科学和数学都很有兴趣。 – She is interested in both science and math.

If you tried 也 in the same position, it would sound wrong:

✕ 她对科学和数学也很有兴趣。 (ungrammatical in this structure)

You could use 也 if you split into two sentences and compared with something mentioned earlier, but not in this exact structure.

Can I say 她很有对科学和数学的兴趣 instead? Is that correct?

她很有对科学和数学的兴趣 is not natural and sounds wrong to native speakers.

The correct and common patterns are:

她对科学和数学很有兴趣。
- Subject + 对 + object + 很有兴趣
她对科学和数学的兴趣很大。
- Subject + 对…的兴趣 + 很大
- Literally: Her interest in science and math is great.

Notice:

In pattern 1, 对 comes right before the thing you’re interested in, and is followed directly by 很有兴趣. You don’t insert 很有 before 对…的兴趣 like in your incorrect version.
In pattern 2, 对科学和数学的兴趣 is a noun phrase (“interest in science and math”), and 很大 is an adjective describing that noun.

So stick to one of the correct patterns:

她对科学和数学都很有兴趣。
她对科学和数学的兴趣都很大。

What is the literal, word‑for‑word breakdown of the whole sentence?

她对科学和数学都很有兴趣。

她 (tā) – she
对 (duì) – towards / with regard to
科学 (kēxué) – science
和 (hé) – and
数学 (shùxué) – mathematics
都 (dōu) – all / both
很 (hěn) – very / quite (degree adverb)
有 (yǒu) – have
兴趣 (xìngqù) – interest

Literal-ish:
She, towards science and math, both, very, has interest.

Natural English:
She is very interested in both science and math.

Are there any tone changes (tone sandhi) I should pay attention to in this sentence?

Yes, there is one common tone sandhi case:

很 (hěn) is 3rd tone.
有 (yǒu) is 3rd tone.

In regular speech, two 3rd tones in a row follow the 3rd–3rd rule:

The first 3rd tone changes to a 2nd‑tone pronunciation.

So 很有 (hěn yǒu) is usually pronounced more like:

héng yǒu (2nd tone then 3rd tone)

You don’t change the written tones, but you pronounce them with this sandhi.

Other words:

对 (duì) – 4th tone
都 (dōu) – 1st tone (here; there is another word dū as in 首都, “capital”, but different character/meaning)

The main sandhi to watch is 很有 → spoken as 2nd + 3rd tone.

Since 他 and 她 are both pronounced tā, does this sentence sound different for “he” and “she”?

No, 他 and 她 are pronounced exactly the same: tā (high-level 1st tone).

So:

她对科学和数学都很有兴趣。 – She is interested in science and math.
他对科学和数学都很有兴趣。 – He is interested in science and math.

These two sentences are pronounced identically. The gender difference exists only in writing, not in speech.

In spoken Mandarin, you rely on context to know whether tā means “he”, “she”, or even “it” (sometimes).