Я так и не поняла, стоит ли отправлять этот запрос сегодня или лучше подождать до утра.

Because the speaker is female. In the past tense, Russian verbs agree with gender and number:

• понял = a man understood / realized
• поняла = a woman understood / realized
• поняло = it understood
• поняли = they understood

So Я так и не поняла means I still didn’t understand / I never figured out, said by a woman.

Так и не is a very common pattern meaning still not, never did, or in the end did not.

So:

• Я не поняла = I didn’t understand
• Я так и не поняла = I still didn’t understand / I never managed to figure out

It adds a feeling that the expected understanding never happened.

You will often see this pattern with perfective verbs:

• так и не понял = never did understand
• так и не смог = never managed to
• так и не пришёл = never came after all

Here ли means whether / if in an indirect question.

The phrase стоит ли... means:

• is it worth ... ?
• should one ... ?
• whether it’s advisable to ...

So стоит ли отправлять этот запрос сегодня means something like:

whether I should send this request today

In Russian, ли usually comes after the word it applies to. That is why it follows стоит:

• стоит ли
• not normally ли стоит

Historically, стоить is related to be worth, so yes, the literal idea is closer to is it worth it than to English should.

But in actual use, стоит ли + infinitive is a very normal way to say:

• should I ... ?
• is it worth ... ?
• would it be better to ... ?

For example:

• Стоит ли ехать? = Should I go? / Is it worth going?
• Стоит ли ждать? = Should I wait? / Is it worth waiting?

So in this sentence it is about uncertainty over the best course of action.

This is a great question, because aspect matters a lot here.

• отправлять is imperfective
• подождать is perfective

Why?

1. стоит ли + imperfective infinitive is very common

After стоит ли, Russian often uses the imperfective infinitive when talking about an action in a general practical sense:

• стоит ли писать
• стоит ли звонить
• стоит ли отправлять

This feels like is it worth doing this action.

2. подождать is naturally perfective

Подождать means to wait for a while / to wait and finish waiting, so it is a bounded, completed action. Here the speaker means or better to wait until morning, a single complete choice.

You could compare:

• лучше подождать до утра = better to wait until morning
• лучше ждать до утра = better to be waiting until morning

The first is the more natural choice here.

Yes, отправить is possible in some contexts, but отправлять sounds very natural here.

Compare the feel:

• стоит ли отправлять этот запрос сегодня = should I send this request today? / is it worth sending it today?
• стоит ли отправить этот запрос сегодня = should I send this request today? with a slightly more one-time, result-focused feel

With стоит ли, Russian often prefers the imperfective when discussing whether doing something is advisable in general. So отправлять is a very idiomatic choice.

Russian often leaves out elements that are obvious from context.

Here, the sentence begins with Я так и не поняла, so it is already clear that the uncertainty belongs to I. Because of that, Russian does not need to repeat мне.

Fuller version:

• Я так и не поняла, стоит ли мне отправлять этот запрос сегодня...

Natural shorter version:

• Я так и не поняла, стоит ли отправлять этот запрос сегодня...

Both are correct. The version without мне is smooth and natural.

Этот means this, and запрос means request, query, or sometimes inquiry, depending on context.

So этот запрос is this request.

Case-wise:

• этот запрос is accusative singular masculine
• it looks the same as nominative because запрос is inanimate

Compare:

• nominative: этот запрос
• accusative: я отправляю этот запрос

Because the preposition до requires the genitive case.

• утро = morning
• genitive: утра

So:

• до утра = until morning

Other examples:

• до вечера = until evening
• до дома = as far as the house / until reaching home
• до понедельника = until Monday

Или means or, and лучше means better.

So the sentence is contrasting two options:

• отправлять этот запрос сегодня
• подождать до утра

The idea is:

whether to send this request today, or whether it would be better to wait until morning

Russian often uses или лучше + infinitive in exactly this way:

• поехать сейчас или лучше остаться дома
• позвонить ей или лучше написать

Because стоит ли отправлять этот запрос сегодня или лучше подождать до утра is a subordinate clause attached to the main clause Я так и не поняла.

Main clause:

• Я так и не поняла

Subordinate indirect question:

• стоит ли отправлять этот запрос сегодня или лучше подождать до утра

Russian normally separates such clauses with a comma.

It is neutral and very natural. It could appear in conversation, email, or ordinary written language.

A few parts influence the tone:

• запрос sounds a bit more formal or professional than everyday words like вопрос or сообщение
• the whole structure is still completely normal in speech

So this could easily be said by someone discussing work:

I still couldn’t figure out whether to send this request today or better wait until morning.

Yes. In Russian, понять often covers both ideas:

• understand
• figure out
• realize

In this sentence, a very natural English rendering is not just I didn’t understand, but:

• I couldn’t figure out
• I still didn’t decide
• I never really understood what to do

That is because the object of understanding is an indirect question: whether to send it today or wait until morning. So the meaning is closer to I couldn’t figure out what the better choice was.