Learn Hardcore Chinese: tā kěyǐ gēn wǒ shuō ma?

Affirmative: 可以。/ 行。/ 没问题。 (Yes / That’s fine / No problem.)
Negative: 不可以。/ 不行。 (No / Not allowed.) You can add a reason: 不可以，她现在很忙。 (No, she’s busy now.)

Does 可以 (kěyǐ) mean permission or ability here? How is it different from 能 (néng) and 会 (huì)?

可以 usually asks about permission or possibility: Is it allowed/OK/possible? That’s the most likely reading here.
能 asks about capability or practical possibility: Is she able to? (e.g., due to rules, time, environment). 她能跟我说吗？ = Is she able/allowed to talk to me?
会 is about learned skill or likelihood/willingness. 她会跟我说吗？ can sound like Will she talk to me? (Is she inclined to?) or Does she know how to talk to me (in X language)?

For skill: 她会说中文吗？ (Does she know how to speak Chinese?)

Can I use 对 or 和 instead of 跟?

跟…说 (talk to/with someone) is very common and colloquial.
对…说 (say to someone) is also correct, often used when quoting speech: 她对我说：“谢谢。”
和 rarely pairs directly with 说; instead use 和…说话 for “talk with”: 她可以和我说话吗？

So your sentence is most natural as 她可以跟我说吗？; 她可以对我说吗？ is also acceptable (slightly more formal/neutral).

Should it be 说 or 说话? What’s the difference?

说 = say/tell (often implies some content). 跟我说 can mean “say/tell me (something).”
说话 = speak/talk (the act of speaking). 跟我说话 emphasizes conversing. If you mean “talk with me” as an activity, 她可以跟我说话吗？ is very natural. If you mean “tell me (something),” 她可以跟我说吗？ or better 她可以告诉我吗？

Is 给我说 correct?

In standard Mandarin, 给我说 is generally not used for “tell me.” Use:

跟我说 (say to me), or
告诉我 (tell me) — the most idiomatic for “tell me.” You might see imperative emphatics like 给我说清楚！ (Explain it clearly to me!), but that’s a different, forceful pattern. For neutral requests, avoid 给我说.

Why is the word order 她 + 可以 + 跟我 + 说 + 吗? Could I say 她跟我可以说吗?

Modal verbs like 可以/能/会 come before the main verb phrase:

Correct: 她可以跟我说吗？
Incorrect/unnatural: 她跟我可以说吗？ General template: Subject + Modal + [Preposition + object] + Verb + (object) + 吗.

Can I ask the same thing without 吗?

Yes; use the A-not-A pattern:

她可不可以跟我说？
她能不能跟我说？ These are very common. A stronger, more pressing version is 她跟我说不说？, which can sound blunt.

How can I make the request softer or more polite?

Add softening elements:

她可以跟我说一下吗？ (…talk to me a bit)
她可以跟我说一会儿吗？ (…for a little while)
她可以跟我说说吗？ (verb reduplication softens) If asking a third party for permission:
可以让她跟我说两句吗？ (Could you let her talk to me for a moment?)

Can I use 讲/讲话 instead of 说/说话?

讲 (jiǎng) and 讲话 (jiǎnghuà) are common in Taiwan/Singapore and parts of southern China.
Mainland Mandarin prefers 说/说话 in everyday conversation. So 她可以跟我讲话吗？ sounds very natural in Taiwan; on the Mainland, 她可以跟我说话吗？ is more neutral.

What’s the difference between 跟我说, 对我说, and 跟我说话?

跟我说: say/tell to me; often implies some content.
对我说: say to me; often used when introducing quoted speech.
跟我说话: talk with me (focus on the act of conversing). For “tell me,” prefer 告诉我. For “talk with me,” 跟我说话 is safest.

How do tones and pronunciation work here?

吗 (ma) is neutral tone.
可以 is written as two third tones (kěyǐ), but in fluent speech the first often surfaces as a rising contour due to 3–3 sandhi: it sounds like kéyǐ.
跟 (gēn) and 说 (shuō) are first tone; 我 (wǒ) is third tone; 她 (tā) is first tone.

How do I answer without saying “yes” or “no”?

Chinese typically repeats the verb/modality:

Positive: 可以。 / 她可以。 / 可以跟你说。
Negative: 不可以。 / 她现在不可以。 You can also answer with 行/不行 or 能/不能 depending on what was asked.

What does 她不可以跟我说吗？ mean? Is that different?

That’s a negative question: “Isn’t she allowed to talk to me?” It can convey surprise or mild challenge. Answers still use 可以/不可以:

What about 了 in 她可以跟我说了吗？?

Adding 了 often signals a change of state or “by now” sense:

她可以跟我说了吗？ = “Is she allowed/able to talk to me now (at this point)?” Useful after waiting for permission or for her to become available.

Can I drop the subject and just say 可以跟我说吗？?

Yes, if context makes the subject clear. Chinese often omits known subjects. Without context, it becomes vague: “Can (someone) talk to me?”

Is 她可以对我说话吗？ acceptable?

It’s understandable but less natural. Prefer:

她可以对我说吗？ (say to me) or
她可以跟我说话吗？ (talk with me) The pairing 对…说话 is uncommon; use 跟…说话 instead.

Any character/orthography notes?

Simplified: 她可以跟我说吗？
Traditional: 她可以跟我說嗎？ Chinese normally doesn’t use spaces between words; they’re shown here only for learning.