El estante está bien sujeto a la pared.

Because estar is commonly used to describe a state or condition, while ser is used more for essential identity or defining characteristics.

Here, está bien sujeto means the shelf is securely attached / is firmly fixed, which describes its current condition. That is why estar is the natural choice.

• El estante está bien sujeto = the shelf is well secured
• El estante es... would sound odd here, because being secured is not treated as its defining identity

Yes, sujeto can mean subject, especially in grammar or in more formal contexts. But in this sentence, it is a past participle/adjective from the verb sujetar, meaning:

• to hold
• to fasten
• to secure
• to attach

So bien sujeto means well secured, firmly attached, or held in place properly.

It is functioning as an adjective here.

In El estante está bien sujeto a la pared, sujeto describes the condition of the shelf: it is secured. This is very common in Spanish: a past participle can be used adjectivally after estar.

Compare:

• La puerta está cerrada = the door is closed
• El cuadro está colgado = the picture is hung
• El estante está sujeto = the shelf is secured

Because bien here means something like properly, securely, or firmly, not just very.

• bien sujeto = properly/securely fastened
• muy sujeto would sound unnatural in this context

With participles and adjectives describing how well something is fixed, bien is very common:

• bien cerrado = properly closed
• bien atado = tied well
• bien sujeto = securely fastened

Because sujetar algo a algo is the normal pattern: to fasten something to something.

So:

• sujetar el estante a la pared = to attach the shelf to the wall
• estar sujeto a la pared = to be attached to the wall

That a is the preposition that links the object to the surface or support it is attached to.

Not in this sentence if you mean attached to the wall.

• a la pared = attached/fastened to the wall
• en la pared = on the wall, referring more to location

So:

• El estante está sujeto a la pared = the shelf is secured to the wall
• El estante está en la pared would not be the normal way to express that idea

If you only want to say where something is, you might use:

• El cuadro está en la pared = the picture is on the wall

But with sujeto, a is the correct preposition.

Estante usually means a shelf: one horizontal board or level.

Estantería usually means a shelving unit, bookcase, or set of shelves.

So in your sentence:

• El estante = the shelf

If it were a whole shelving structure, you might say:

• La estantería está bien sujeta a la pared = the shelving unit is securely attached to the wall

Sometimes, but they are not all equally natural.

• sujeto = secured, held in place
• fijado = fixed, firmly attached
• agarrado = grabbed/held; in some contexts it can work, but it is less precise here
• atado = tied, so it suggests using rope/string or something similar

For a shelf attached to a wall, the most natural options are:

• está bien sujeto a la pared
• está bien fijado a la pared

Atado would usually sound wrong unless it is literally tied.

Because it agrees with el estante, which is masculine singular.

Agreement in Spanish is important:

• El estante está bien sujeto
• La balda está bien sujeta
• Los estantes están bien sujetos
• Las baldas están bien sujetas

So the ending changes depending on the noun being described.

It is a correct and common word, but in Spain you will also hear balda for a shelf, especially when talking about one individual shelf inside a bookcase or cupboard.

Very roughly:

• estante = shelf
• balda = shelf/plank, often very natural in Spain
• estantería = shelving unit/bookcase

So in Spain, someone might also say:

• La balda está bien sujeta a la pared

But El estante está bien sujeto a la pared is perfectly correct.

Yes. Sujeto a can also mean subject to, dependent on, or liable to, especially in formal language.

For example:

• El pago está sujeto a condiciones. = The payment is subject to conditions.
• Los cambios están sujetos a aprobación. = The changes are subject to approval.

So the phrase has both:

1. a physical meaning: attached/secured to
2. a figurative/formal meaning: subject to

In your sentence, it is clearly the physical meaning.

Yes, both are possible.

• El estante está sujeto a la pared = the shelf is attached to the wall
• El estante está bien sujeto a la pared = the shelf is securely/properly attached to the wall

Adding bien emphasizes that it is attached firmly or correctly, not loosely or badly.

Yes. The sentence describes the resulting state of the shelf, not the action of attaching it.

• El estante está bien sujeto a la pared = the shelf is in a secured state
• Han sujetado el estante a la pared = they have attached the shelf to the wall
• Sujetaron el estante a la pared = they attached the shelf to the wall

So estar + participle/adjective focuses on the condition, while a normal verb form focuses on the action.