No estoy lo suficientemente despierto como para empezar otra redacción ahora.

Because despierto here describes a temporary state: being awake or alert right now.

• estar despierto = to be awake / alert
• ser is generally used for more permanent or defining characteristics

So:

• Estoy despierto = I’m awake
• Soy despierto can mean something quite different, more like I’m clever / quick-witted

In this sentence, the speaker is talking about their current level of alertness, so estar is the correct verb.

Lo suficientemente means enough or sufficiently, especially before an adjective or adverb.

In this sentence:

• lo suficientemente despierto = awake enough

The lo here is not the and not a direct object. It is a neuter word used to turn the adjective phrase into something like to a sufficient degree.

A useful pattern is:

• lo suficientemente + adjective/adverb + como para...

Examples:

• No es lo suficientemente alto = He/She isn’t tall enough
• Corres lo suficientemente rápido = You run fast enough

English speakers often want a word-for-word match, but it’s best to learn lo suficientemente as a set phrase.

This is a very common Spanish pattern for expressing enough ... to ...

The structure is:

• lo suficientemente + adjective/adverb + como para + infinitive

So here:

• lo suficientemente despierto = awake enough
• como para empezar... = to start...

Put together:

• No estoy lo suficientemente despierto como para empezar...
  = I’m not awake enough to start...

Other examples:

• No estoy lo suficientemente cansado como para irme a dormir = I’m not tired enough to go to sleep
• Es lo suficientemente fuerte como para levantarlo = He/She is strong enough to lift it

Because para + infinitive is used when the subject of both actions is the same.

Here, the person who isn’t awake enough is also the person who would start another essay. So Spanish uses the infinitive:

• No estoy... como para empezar...

If the subject changed, Spanish would normally use para que + subjunctive.

Compare:

• No estoy lo suficientemente despierto como para empezar otra redacción
  = I am not awake enough to start another essay

• No estoy lo suficientemente despierto para que me expliques eso ahora
  = I’m not awake enough for you to explain that to me now

So:

• same subject → para + infinitive
• different subject → para que + subjunctive

Because despierto is the normal adjective meaning awake or alert.

• estar despierto = to be awake

Despertado is the past participle of despertar, but it is not usually the natural choice for this meaning. In most everyday Spanish, if you want to say someone is awake, you say:

• Está despierto

not usually:

• Está despertado

So in this sentence, despierto is the standard and natural adjective.

Yes. It must agree with the person in gender and number if Spanish requires that distinction.

Forms:

• despierto = masculine singular
• despierta = feminine singular
• despiertos = masculine or mixed plural
• despiertas = feminine plural

Examples:

• No estoy lo suficientemente despierto... = said by a man
• No estoy lo suficientemente despierta... = said by a woman
• No estamos lo suficientemente despiertos... = said by a group including at least one male, or in a mixed group
• No estamos lo suficientemente despiertas... = said by a group of women

Yes, that is possible, and native speakers will understand it. But lo suficientemente... como para... is a very common and natural full structure.

Compare:

• No estoy suficientemente despierto = I’m not sufficiently awake
• No estoy lo suficientemente despierto como para empezar... = I’m not awake enough to start...

Using lo often sounds more complete and idiomatic, especially when followed by como para.

Also, Spanish has another very common alternative:

• No estoy bastante despierto como para...

That also means I’m not awake enough to...

Here redacción most likely means essay, composition, or writing assignment.

In school contexts, una redacción often means a piece of written work, especially something like a composition for class.

Depending on context, redacción can also mean:

• the act of writing something
• the wording of a text
• a newsroom/editorial office

But in this sentence, because of empezar otra redacción, the natural meaning is start another essay/composition.

Because otro / otra normally replaces the indefinite article in Spanish.

So Spanish says:

• otra redacción = another essay

not:

• una otra redacción

This is similar to:

• otro día = another day
• otra vez = another time / again
• otro libro = another book

Yes, Spanish word order is flexible, especially with time words like ahora.

The original sentence:

• No estoy lo suficientemente despierto como para empezar otra redacción ahora.

You could also say:

• Ahora no estoy lo suficientemente despierto como para empezar otra redacción.
• No estoy ahora lo suficientemente despierto como para empezar otra redacción.
  (less common, but possible depending on emphasis)

The original version sounds very natural. Placing ahora at the end often gives a smooth, conversational rhythm.

Because Spanish often leaves out subject pronouns when the verb already makes the subject clear.

• estoy already tells us the subject is I

So:

• No estoy lo suficientemente despierto... = perfectly natural
• Yo no estoy lo suficientemente despierto... = also correct, but more emphatic

You might include yo if you want contrast or emphasis, for example:

• Yo no estoy lo suficientemente despierto, pero ella sí.
  = I’m not awake enough, but she is.

Empezar is completely natural here and means to start or to begin.

You could also use:

• comenzar = to begin
• ponerse a escribir = to get down to writing / start writing

Examples:

• No estoy lo suficientemente despierto como para comenzar otra redacción ahora.
• No estoy lo suficientemente despierto como para ponerme a escribir otra redacción ahora.

These are all correct, but empezar is simple, common, and very natural.

Yes. A few natural alternatives are:

• No estoy lo bastante despierto como para empezar otra redacción ahora.
• No estoy suficientemente despierto para empezar otra redacción ahora.
• No tengo la cabeza como para empezar otra redacción ahora.
  This is more idiomatic and means something like I’m not in the right frame of mind to start another essay now.

The original sentence is already very natural and clear. The main thing to recognize is the pattern:

• No estoy lo suficientemente + adjective + como para + infinitive