Learn Hardcore Russian: Чтобы поддержать мою соседку, я сделал ей комплимент и сказал, что она сегодня выглядит лучше. - To support my neighbor, I gave her a compliment and said she looks better today.

Чтобы + infinitive introduces a purpose clause: in order to / so as to do something.
So Чтобы поддержать мою соседку, ... literally sets up the goal/purpose, and then the main clause explains what you did.

Why is there a comma after соседку: Чтобы поддержать мою соседку, я...?

In Russian, an introductory purpose clause with чтобы is normally separated from the main clause by a comma.
Structure:

Purpose clause: Чтобы поддержать мою соседку
Main clause: я сделал... и сказал...

Why is it поддержать (perfective) and not поддерживать (imperfective)?

With чтобы, Russian often uses the perfective infinitive to express a single, intended result: to support (successfully / as an action with an endpoint).

Чтобы поддержать = in order to give support (as a concrete act)
Чтобы поддерживать would sound more like ongoing/regular support: in order to keep supporting her over time.

Why is it мою соседку (accusative) and not моей соседке (dative)?

The verb поддержать takes a direct object in the accusative: поддержать кого? что?
So: поддержать (мою) соседку.
Using dative (моей соседке) would fit different verbs/expressions (like помочь соседке), but not the standard government of поддержать.

What case is ей, and why is it used in я сделал ей комплимент?

Ей is dative (to her). The pattern is:

сделать комплимент кому? = to give/pay a compliment to someone (dative)
So ей marks the recipient of the compliment.

Is сделал комплимент the normal way to say gave a compliment? Why not сказал комплимент?

Yes, сделать комплимент is a very common collocation (idiomatic pairing).
You can also say сказать комплимент, but it often sounds less idiomatic or more “literal.” Many speakers prefer сделать here.

Why is there no comma before и сказал?

Because я сделал... и сказал... is a compound predicate with one subject (я) and two verbs joined by и. Russian normally does not put a comma between two verbs like that when they share the same subject and form one sequence of actions.

Why is there a comma before что: сказал, что...?

Because что она сегодня выглядит лучше is a subordinate clause (object/content clause) introduced by что. Russian separates such subordinate clauses with a comma:
сказать (что?), so the что-clause functions as what you said.

Why do we need что at all? Can it be omitted?

In Russian, after verbs like сказать, you typically introduce reported content with что. Omitting it is possible in very colloquial speech, but in neutral/written Russian it’s expected:

Neutral: сказал, что она выглядит лучше
Colloquial sometimes: сказал она выглядит лучше (often considered incorrect or too informal)

Does она сегодня выглядит лучше mean better than yesterday, or better than usual? Why is there no чем?

By itself, выглядит лучше means looks better with an implied comparison (better than before / better than usual), depending on context.
Russian can leave the comparison “open” without чем.... If you want to specify explicitly, you can add:

лучше, чем обычно = better than usual
лучше, чем вчера = better than yesterday

What does выглядит literally mean, and why is it present tense when the rest is past?

выглядит = (she) looks / appears (from выглядеть).
It’s present tense because it’s inside reported speech/content: you said that at that time she looks better (a statement presented as currently true in the moment of speaking). Russian commonly keeps present tense in such reported statements.

Does сказал tell us anything about the speaker?

Yes. сказал is past tense masculine singular, so it implies the speaker (я) is male.
If the speaker is female, it would be сказала:
Чтобы поддержать мою соседку, я сделала ей комплимент и сказала, что она сегодня выглядит лучше.