Learn Hardcore Chinese: nǐ kěyǐ gěi tā dǎ diànhuà, ràng tā fàngxīn.

可以: permission/possibility/suggestion.
能 (néng): ability or feasibility given conditions.
会 (huì): learned skill or likelihood (will). Examples: 你能给她打个电话吗？ (Are you able to call her?) 你会给她打电话吗？ (Will you call her?/Do you tend to call her?)

Why is 给 (gěi) placed before 打电话 (dǎ diànhuà)? Can I say 打电话给她?

给 here is a coverb meaning to or for, marking the recipient before the main action: 给她打电话. In Taiwan and some southern varieties, 打电话给她 is also common. Both are fine in Mandarin; mainland speech prefers 给她打电话.

Can I use 和, 对, or 跟 with 打电话 to mark the person?

Normally, no. Use 给: 给她打电话 or 打电话给她. If you want the idea of being on the phone with someone, say 跟/和她通电话. For speaking to someone (not specifically by phone), use 对/跟她说.

Why is 打 (dǎ) used with 电话? Does it mean hit?

打 is a light verb used in many set phrases to mean perform/do: 打电话 (make a phone call), 打字 (type), 打球 (play ball), 打招呼 (greet). It doesn’t carry the literal hit meaning here.

What’s the difference between 打电话 and 打个电话?

打电话 names the activity in general. Inserting a measure word makes it one instance: 打个电话 (make a call). In suggestions, 打个电话 often sounds more natural: 你可以给她打个电话.

Why is 她 (tā) repeated twice? Can I drop the second 她 in 让她放心?

You can’t drop it. 让 is a causative verb that needs an explicit object: you cause her to feel at ease. 让放心 without 她 would be ungrammatical; keep 让她放心.

What does 让 (ràng) mean here? How is it different from 叫 (jiào) and 使 (shǐ)?

Here 让 means cause/let/make: 让她放心 = make her feel reassured. 叫 can also mean make/let (more colloquial/regional). 使 is more formal/literary. In everyday speech, 让 is the default.

What exactly does 放心 (fàngxīn) mean? Is there a negative or opposite?

放心 means to stop worrying, to feel reassured. Opposite/contrast: 担心 (to worry). Useful patterns:

我放心了 (I’m relieved now).
我对他不放心 (I don’t feel at ease about him).
Imperative soothing: 放心吧 (Don’t worry).

What function does the comma have? Is it like so that?

Yes. The second clause is a purpose/result clause: do A, so that B. You can make this explicit with:

好让她放心 (so that she can be at ease)
这样她就放心了 (this way she’ll be reassured)

Is this a question or a suggestion? How do I make it a polite request?

As written, it’s a neutral suggestion/statement. To ask politely:

你可以给她打个电话，让她放心吗？
你给她打个电话，让她放心吧。
要不你给她打个电话，让她放心？

Where would 了 (le) go if the call already happened or to show the expected result?

Completed action: 你给她打了电话.
Result achieved: 这样她就放心了.
Both: 你给她打了个电话，这样她就放心了。

How do I pronounce the string of third tones here with tone sandhi?

In fluent speech you’ll typically get: ní ké yí gěi tā dǎ diànhuà, ràng tā fàngxīn. Explanation: in chains of 3rd tones, all but the last become 2nd. So 你(3) 可以(3-3) 给(3) surfaces roughly as ní ké yí gěi.

Does 她 necessarily mean a female in speech?

In speech 他/她/它 are all pronounced tā. The written form 她 specifies female; in conversation, gender is inferred from context. For mixed or unknown groups, 他们 (tāmen) is the default in writing.

Any small tweaks a native might add to sound more natural?

Common touches:

Add a measure word: 你可以给她打个电话.
Soften the result: 让她放心一下/放心点儿.
Add a suggestion particle: …吧.
Link result explicitly: 你给她打个电话吧，这样她就放心了。

Can I omit the subject 你?

Yes, if context makes it clear who should act: 可以给她打个电话，让她放心。 This sounds less direct and more like a general suggestion; include 你 if you’re addressing someone specifically.

How do I say call her back or get through?

Call back: 给她回电话/回个电话.
Get through: 打通了; can’t get through: 打不通. Example: 有空给她回个电话，让她放心。

Is 给她一个电话 okay to mean give her a call?

Colloquially you may hear 给我一个电话 meaning call me, but it can be ambiguous or sound like give me a phone/number. Clearer is 给她打个电话 or 给她来个电话 (colloquial).

Can I use 能 or 会 instead of 可以 here?

You can, but the nuance changes:

能 stresses ability/feasibility: 你能给她打个电话吗？
会 can mean will/likely to: 你会给她打电话吗？ (Will you?)
可以 is neutral permission/suggestion: 你可以给她打个电话…