Mi perro es obediente.

• Mi = my. It is a possessive adjective.
• perro = dog.
• es = is. It is the third-person singular form of ser.
• obediente = obedient.

So the structure is:

Mi perro (subject) + es (verb) + obediente (adjective)

Use mi before a noun:

• mi perro = my dog

Use mío / mía / míos / mías when the noun is not stated or when it is used more like mine:

• El perro es mío = The dog is mine

So in Mi perro es obediente, you need mi because it comes directly before perro.

In Spanish, a possessive like mi usually replaces the article.

So Spanish says:

• mi perro

not:

• el mi perro
• un mi perro

This is different from English, but it is the normal pattern in Spanish.

Spanish usually uses ser for characteristics that are seen as more general or defining, and estar for states or conditions.

• Mi perro es obediente = My dog is obedient.
  This describes obedience as a trait or characteristic.

If you say:

• Mi perro está obediente

that can sound more like My dog is being obedient right now or My dog is acting obedient at the moment.

So es is the most natural choice if you mean obedience as part of the dog's character.

Yes, but the meaning changes a little.

• Mi perro es obediente = my dog is obedient in general
• Mi perro está obediente = my dog is being obedient right now / at the moment

So both are possible, but they are not exactly the same.

Some Spanish adjectives change form for gender, and some do not.

• alto / alta
• bonito / bonita

But obediente is an adjective ending in -e, so it usually stays the same for both masculine and feminine singular:

• Mi perro es obediente
• Mi perra es obediente

It does change for number:

• Mis perros son obedientes
• Mis perras son obedientes

Then you would normally say:

• Mi perra es obediente

Here:

• perra = female dog
• obediente stays the same, because it does not change for gender in the singular

In this sentence, obediente is not directly attached to the noun. It comes after es because it is part of a predicate:

• Mi perro = subject
• es obediente = what you say about the subject

This is very normal in Spanish, just like in English:

• My dog is obedient

If you put an adjective directly before or after a noun in Spanish, that is a different structure.

Because mi does not change for gender in the singular.

• mi perro = my dog
• mi perra = my female dog
• mi casa = my house

It does change for number:

• mis perros
• mis casas

So the change is singular vs. plural, not masculine vs. feminine.

You would change the possessive, the noun, the verb, and the adjective as needed:

• Mis perros son obedientes = My dogs are obedient

Changes:

• mi → mis
• perro → perros
• es → son
• obediente → obedientes

The normal order is:

• Mi perro es obediente

You can change the order in special contexts for emphasis, but that is less neutral:

• Obediente es mi perro
• Mi perro obediente... (not the same structure)

For a basic statement, Mi perro es obediente is the standard and most natural order.

A simple Latin American-style pronunciation guide:

• Mi ≈ mee
• perro ≈ PEH-rroh
  
  • The rr is a rolled or trilled r
• es ≈ ess
• obediente ≈ oh-beh-dee-EN-teh

A rough full version:

mee PEH-rroh ess oh-beh-dee-EN-teh

Obediente is completely correct and natural.

Native speakers might also use other words depending on nuance:

• bien portado = well-behaved
• disciplinado = disciplined
• tranquilo = calm

But Mi perro es obediente is a very normal and useful sentence.