Este taburete es firme y cómodo.

Because taburete is a masculine singular noun in Spanish.
The demonstrative este must agree with the noun it describes:

• este = this, for masculine singular nouns
• esta = this, for feminine singular nouns

So:

• este taburete = this stool
• esta silla = this chair

Not quite. Taburete usually means stool, especially a stool without a backrest, or a bar stool.
Silla means chair.

So taburete refers to a different kind of seat from silla.

Es is the third person singular form of the verb ser:

• ser = to be
• es = is

It is used because the subject is este taburete = this stool, which is singular.

So:

• Este taburete es... = This stool is...

Spanish has two verbs for to be: ser and estar.

Here we use ser because firme and cómodo describe qualities or characteristics of the stool. They are presented as inherent or defining features.

• es firme y cómodo = it is firm and comfortable

If you used estar, it would sound unusual here, because estar is more often used for states, conditions, or location, not general product qualities.

In Spanish, adjectives often come after the noun, especially when they describe ordinary qualities.

So:

• taburete firme y cómodo / Este taburete es firme y cómodo

This is normal Spanish word order.
In English, adjectives usually go before the noun: a firm, comfortable stool.
But in Spanish, after ser, adjectives naturally come after the verb:

• es firme y cómodo

Because este already works as a demonstrative determiner: this.

In Spanish, you normally do not say:

• este el taburete ❌

You say:

• este taburete = this stool

Just like in English you say this stool, not this the stool.

The written accent in cómodo shows where the stress falls: CÓ-mo-do.

Without the accent, Spanish spelling rules would suggest a different stress pattern.
The accent mark helps show the correct pronunciation.

Also, this accent mark is part of the normal spelling of the word, so it should always be written:

• cómodo ✅
• comodо ❌

Because cómodo must agree with taburete, which is masculine singular.

Agreement works like this:

• taburete cómodo = comfortable stool
• silla cómoda = comfortable chair

So the adjective changes depending on the gender and number of the noun.

Because firme is an adjective with the same form for masculine and feminine singular.

Compare:

• un taburete firme
• una silla firme

But in the plural it changes:

• taburetes firmes
• sillas firmes

So firme does agree in number, but not in gender in the singular.

Yes. That is grammatically correct.

The order of adjectives linked by y can often be changed without changing the basic meaning:

• firme y cómodo
• cómodo y firme

Sometimes the order just reflects what the speaker wants to emphasize first.
For example, a seller might say firme y cómodo to stress stability first.

Y means and.

So:

• firme y cómodo = firm and comfortable

Usually, yes, y is the normal word for and in Spanish.
One small exception: before words beginning with an i sound, Spanish often uses e instead of y for smoother pronunciation.

Example:

• padre e hijo = father and son

But in your sentence, y is correct.

It would become:

• Estos taburetes son firmes y cómodos.

Here is what changes:

• este → estos because the noun is now plural
• taburete → taburetes
• es → son
• firme → firmes
• cómodo → cómodos

Everything that needs to agree with the plural noun changes accordingly.

A simple pronunciation guide is:

EH-steh tah-boo-REH-teh ess FEER-meh ee KOH-moh-doh

A few useful notes for Spain Spanish:

• e is usually like eh
• u is like oo
• r in firme is a light tap
• y here is pronounced like ee
• stress falls on:
  • EH-ste
  • tabu-RE-te
  • FEER-me
  • CÓ-mo-do

If you want a more Spain-specific pronunciation, z and soft c can sound like th, but this sentence does not contain that sound.