Хоть дедлайн и близко, я всё равно хочу сначала перечитать эссе ещё раз.

It is a very common concessive pattern, meaning something like:

Even though ..., I still ...
or
Although ..., I still ...

So here:

Хоть дедлайн и близко, я всё равно хочу...
= Even though the deadline is close, I still want...

A useful point: хоть here is similar to хотя, but хоть sounds a bit more conversational.

Here и does not mean and.

In sentences with хоть / хотя, Russian often uses и to strengthen the contrast:

Хоть X и Y, ...
= Although X is Y, ...

So:

Хоть дедлайн и близко...
means Although the deadline is close...

This и is a particle here, not a coordinating conjunction. It often sounds very natural in this structure, though in some cases it can be omitted.

These are three different forms with different jobs:

• близкий = full adjective, used before a noun
  
  • близкий дедлайн = a near/close deadline
• близок = short-form adjective, used as a predicate
  
  • дедлайн близок = the deadline is near
• близко = a predicative/adverb-like form, also used to mean near / close / soon

In this sentence, близко works as the predicate:
дедлайн близко = the deadline is close / near

This is quite natural in conversational Russian, especially with time-related ideas.
A somewhat more formal or bookish version might be дедлайн близок.

Because in the present tense, Russian normally leaves out the verb to be.

So where English says:

The deadline is close

Russian simply says:

Дедлайн близко

You do not normally insert есть here.

This is one of the most basic differences between English and Russian sentence structure.

Here всё равно means:

• still
• anyway
• all the same

So:

я всё равно хочу...
= I still want... / I want to anyway...

It shows that the speaker’s intention does not change, despite the situation mentioned before.

So the logic is:

• the deadline is close
• but despite that
• I still want to reread the essay first

Сначала here means first.

So:

я всё равно хочу сначала перечитать эссе...
= I still want to reread the essay first...

It tells you about the order of actions: before doing something else, the speaker wants to reread the essay.

That is different from simply saying:

я хочу перечитать эссе
= I want to reread the essay

With сначала, the speaker implies there may be another action after that, such as editing it, sending it, or submitting it.

Перечитать means to read through again / reread.

The prefix пере- often gives the idea of doing something again. So yes, перечитать already includes the idea of repetition.

That means ещё раз is not strictly necessary. The speaker could say:

хочу сначала перечитать эссе

But adding ещё раз is very natural. It makes the repetition extra explicit, a bit like English:

I want to reread the essay one more time

So this combination is not wrong or strange; it is just a bit more emphatic.

Also, перечитать is perfective, which fits well after хочу when the speaker means a single completed action:

• хочу перечитать = I want to reread it (and finish that rereading)

If you used the imperfective перечитывать, it would sound more like an ongoing, repeated, or process-focused action, which is not the main idea here.

It is the direct object here, so logically it is in the accusative.

But эссе is one of those Russian loanwords that is usually indeclinable, meaning its form does not change across cases.

So:

• nominative: эссе
• accusative: эссе
• after many prepositions: still эссе

That is why after перечитать you still see:

перечитать эссе

and not a changed ending.

Дедлайн is a very common modern borrowing, especially in:

• work contexts
• study
• business
• online communication

So this sentence sounds natural in contemporary Russian, especially in casual or semi-professional speech.

A more native-Russian alternative would be something like:

• срок
• срок сдачи

For example:

Хотя срок сдачи уже близок, я всё равно хочу сначала перечитать эссе ещё раз.

That sounds a bit more formal or neutral.
Дедлайн sounds more modern and conversational.