Hoy quiero hacerme una trenza sencilla y ponerme la diadema azul.

Because me is attached to an infinitive.

• hacerme = hacer + me
• ponerme = poner + me

In Spanish, when you have an infinitive like hacer or poner, a pronoun can be attached to the end:

• hacerme = to do/make myself
• ponerme = to put on myself

This is very common after verbs like querer, ir a, poder, etc.

Here me means to myself / on myself.

So:

• hacerme una trenza = to do myself a braid
• ponerme la diadema = to put the headband on myself

In natural English, you usually would not say myself, but Spanish often uses this pronoun where English does not.

Because both actions are being done to the speaker.

• hacerme una trenza = I do a braid for myself
• ponerme la diadema = I put the headband on myself

If you remove me, the meaning changes or becomes less specific:

• hacer una trenza = to make a braid
• poner la diadema = to put the headband

That could be for someone else, or just describe the action more generally.

Yes. That is also correct.

With querer + infinitive, the pronoun can usually go:

• before the conjugated verb: me quiero hacer
• attached to the infinitive: quiero hacerme

So both are possible:

• Hoy quiero hacerme una trenza sencilla
• Hoy me quiero hacer una trenza sencilla

In your full sentence, the original version is very natural because each pronoun stays with its infinitive:

• Hoy quiero hacerme una trenza sencilla y ponerme la diadema azul.

Because querer is followed directly by an infinitive in Spanish.

• quiero hacerme = I want to do myself...
• quiero ponerme = I want to put on...

There is no extra word like English to in this structure.
So do not say quiero a hacerme here.

Both are possible, but they focus slightly differently.

• hacerme una trenza = to do myself a braid / to put my hair into a braid
• trenzarme el pelo = to braid my hair

The first one focuses more on the resulting hairstyle: a braid.
The second focuses more on the action done to the hair.

In Spain, hacerse una trenza sounds very natural.

Because in Spanish, descriptive adjectives often come after the noun.

• una trenza sencilla = a simple braid
• una diadema azul = a blue headband

That is the normal order. Putting sencilla before the noun would sound more marked, stylistic, or subjective.

So una trenza sencilla is the neutral, everyday phrasing.

Here sencilla means simple, plain, or not elaborate.

So una trenza sencilla suggests a braid that is:

• easy
• basic
• not fancy
• not complicated

It does not mean bad or boring. It just describes the style as uncomplicated.

Because la suggests a specific blue headband.

• la diadema azul = the blue headband
• una diadema azul = a blue headband

Using la usually means the speaker already has a particular one in mind, perhaps one they own or one already known in the conversation.

So this sentence sounds like: there is a specific blue headband I want to wear.

In everyday Spanish from Spain, diadema usually means headband or hairband worn on the head.

Depending on context, it can also mean diadem or something crown-like, but in normal daily speech in Spain, the most likely meaning here is headband.

So in this sentence, it is almost certainly a hair accessory.

Because color adjectives normally go after the noun in Spanish.

• la diadema azul = the blue headband
• el vestido rojo = the red dress
• la camisa blanca = the white shirt

This is the standard word order. Putting the color before the noun is unusual in normal everyday speech.

Because Spanish often uses the present tense to talk about a current intention, desire, or plan.

• Hoy quiero hacerme una trenza... = Today I want / feel like doing a braid...
• Hoy quiero ponerme la diadema azul = Today I want / feel like wearing the blue headband

This sounds natural because the wanting is happening now, even if the action happens later today.

If you wanted to sound more like a plan, you could also say:

• Hoy voy a hacerme una trenza sencilla...
• Hoy me haré una trenza sencilla...

But quiero is perfectly natural here.

Yes, but the meaning or emphasis may shift a little.

• Hoy quiero hacerme una trenza sencilla... = emphasis on today
• Quiero hacerme hoy una trenza sencilla... = also possible, but less neutral
• Quiero hacerme una trenza sencilla hoy... = possible, often more conversational

Starting with hoy is very natural because it sets the time frame immediately.

Yes. Una trenza means one braid.

If you wanted more than one, you would change it:

• dos trenzas = two braids
• hacerme unas trenzas = do some braids for myself

So this sentence specifically describes a hairstyle with one braid.