Learn Hardcore Chinese: tā yǐjīng gěi wǒ tā de diànnǎo le.

By default, yes: the first 她 is the subject (she), and the second 她的 means her (own) computer. If you want to make it crystal clear that it’s the subject’s own computer, add 自己:

她已经把自己的电脑给我了。 = She has already given me her own computer. If you meant another woman’s computer, context or a name is used:
小王已经把她的电脑给我了。 (The possessor is some other “she” known from context.)

Is 已经 necessary when there is 了? Can I drop one of them?

了 marks a completed event or new situation.
已经 adds the sense of “already,” emphasizing timing. You can use either alone, but they’re often used together:
她给我她的电脑了。 (Completed event; no emphasis on “already.”)
她已经给我她的电脑了。 (Completed event, with “already.”)
Using 已经 without some completion/result marker often feels unfinished if it’s a standalone sentence: 她已经给我她的电脑 (sounds incomplete unless followed by more). So keep 了 or add another result.

Can I drop 了 here?

Not if you want to assert the transfer has been completed. Without 了, the sentence can sound incomplete or like background info requiring continuation. Acceptable alternatives without sentence-final 了 include:

Use verb-了: 她已经给了我她的电脑。
Use a resultative or clear time marker: 她已经把她的电脑给我了 (still ends with 了), or add context like …，所以我现在在用。

Why is 了 at the end instead of after 给? What’s the difference between 给了 and sentence-final 了?

给了: verb-了 marking completion of the action itself.
Sentence-final 了: marks a new situation/result now holds. Both are fine, and you may see either:
她已经给了我她的电脑。
她已经给我她的电脑了。 In speech, “double 了” is common and natural: 她已经给了我她的电脑了。 In careful writing, many people choose just one.

Can/should I use 把 here?

Yes. 把 is very natural when the direct object is long/specific and you want to foreground it:

她已经把她的电脑给我了。
她已经把她的电脑给了我。 These are often preferred for smoothness when both a recipient and a specific object are present.

Could I use 过 instead of 了?

Not for this meaning. 过 marks experiential “have ever done,” not a specific completed event with a current relevance/state:

她给过我她的电脑。 = She has given me her computer before (at some point); it doesn’t say anything about the current ownership/state. To report the completed transfer as a new fact, use 了 (and optionally 已经).

What is 给 doing here—verb or preposition? What are the common patterns?

Here 给 is a verb meaning “to give,” and it takes two objects:

Pattern A (ditransitive): S + 给 + Recipient + Thing
Example: 她给我她的电脑了。
Pattern B (with 把): S + 把 + Thing + 给 + Recipient
Example: 她把她的电脑给我了。
With verb-了: 她给了我她的电脑。 All are common; choose based on emphasis and flow.

Is the word order 她已经给她的电脑我了 okay?

No. You can’t place the thing before the recipient in this pattern. Use:

她已经给我她的电脑了。
她已经给了我她的电脑。
她已经把她的电脑给我了。 Note: 给我了她的电脑 is heard colloquially in some regions, but the more standard and widely accepted order is 给了我她的电脑.

Do I need a measure word with 电脑?

With a specific possessed noun like 她的电脑, no extra measure is needed.
With an indefinite noun, you must use a measure word, typically 台:
- 她给了我一台电脑。 = She gave me a computer.

Can I omit 的 in 她的电脑?

Not here. With non-kinship/natural-part nouns like 电脑, you need 的: 她的电脑.
Omitting 的 is common only with close kinship terms and a few inalienable relationships: 她妈妈, 我同学. For objects, keep 的.

Does 给 mean “gift” here? How do I say “gave as a gift” or “lent/handed over”?

给 itself is neutral: it can be giving, handing over, or transferring.

Gift: 送给 (optionally with 把)
她已经把她的电脑送给我了。
Lend: 借给
她已经把她的电脑借给我了。
Hand over (physically): 交给
她已经把她的电脑交给我了。 Use the verb that matches your intended nuance.

How do I say the negative “She hasn’t given me her computer yet”?

Use 还没(有) and drop 了:

她还没把她的电脑给我。
她还没有把她的电脑给我。
她还没给我她的电脑。
Using 已经 with a negative here is not idiomatic; use 还没(有).

Any pronunciation/tone-sandhi tips in this sentence?

给 gěi (3rd tone) + 我 wǒ (3rd tone): the first third tone becomes a rising second in speech: sounds like géi wǒ.
了 le is neutral tone.
已经 yǐjīng: yǐ is a “half-third” in fluent speech; jīng is high level (1st).
电脑 diànnǎo: 4th then 3rd (the second often realized as a low “half-third” in flow).

How does Chinese express the English present perfect “has already given”?

Chinese doesn’t mark tense the same way. The combo of aspect and adverbs does the job:

Completion/new situation: 了
“Already”: 已经 So 她已经给我她的电脑了 naturally covers “She has already given me her computer.”

Can I topicalize the object for emphasis?

Yes. Front the object (especially with something long/specific):

她的电脑，她已经给我了。 This highlights “her computer” as the topic, then comments that it has already been given to you.

Is the original sentence natural, or should I prefer a 把 or 给了 version?

The original 她已经给我她的电脑了 is grammatical and acceptable. In everyday speech and writing, many speakers prefer:

她已经把她的电脑给我了。
她已经给了我她的电脑。 These often flow a bit more smoothly when you have both a recipient and a specific object.