Mi vecino se siente bienvenido.

Because the verb here is sentirse, not just sentir.

Spanish usually uses sentirse when someone is describing their own emotional or personal state:

• me siento = I feel
• te sientes = you feel
• se siente = he/she feels

So Mi vecino se siente bienvenido uses se because mi vecino is a third-person singular subject.

Without se, sentir usually means to feel/sense something:

• Siento dolor. = I feel pain.
• Siente miedo. = He feels fear.

Because sentir is a stem-changing verb in the present tense. Its e changes to ie in most forms.

Present tense of sentirse:

• me siento
• te sientes
• se siente
• nos sentimos
• se sienten

So with mi vecino, the correct form is se siente.

In many voseo areas of Latin America, you may also hear:

• vos te sentís

Because bienvenido agrees with vecino, which is masculine singular.

In Spanish, adjectives usually match the noun they describe in gender and number:

• Mi vecino se siente bienvenido.
• Mi vecina se siente bienvenida.
• Mis vecinos se sienten bienvenidos.
• Mis vecinas se sienten bienvenidas.

It agrees with the neighbor, not with the speaker.

Usually, yes.

In a sentence like Mi vecino, most learners should understand it as my male neighbor. If the neighbor is female, Spanish normally uses vecina:

• Mi vecina se siente bienvenida.

In some broader contexts, masculine forms can be used generically, but in a singular sentence like this, vecino is normally read as masculine.

Here, it works as an adjective.

Even though bienvenido looks like a past participle, in modern Spanish you should think of it mainly as the adjective welcome in sentences like this:

• sentirse bienvenido = to feel welcome
• ser bienvenido = to be welcome
• hacer sentir bienvenido = to make someone feel welcome

So in this sentence, bienvenido describes how the neighbor feels.

They are similar, but the focus is different:

• se siente bienvenido = he feels welcome
  
  • This is about his personal experience or perception.
• es bienvenido = he is welcome
  
  • This is a more general statement about how others receive or accept him.

So se siente bienvenido is more subjective.

Because mi already functions as a determiner.

In Spanish, possessives like mi, tu, su, nuestro usually replace the article:

• mi vecino = my neighbor
• tu casa = your house
• su trabajo = his/her/their job

So el mi vecino is not standard modern Spanish.

No, not for this meaning.

• Mi vecino siente bienvenido is not grammatical here.
• Mi vecino está bienvenido is not idiomatic Spanish.

For this idea, Spanish normally uses:

• Mi vecino se siente bienvenido.
• Mi vecino es bienvenido.

Use se siente bienvenido when you want to emphasize how he experiences the situation.

You keep the pattern sentirse + adjective, but both the reflexive pronoun and the adjective must fit the subject.

Examples:

• Me siento bienvenido / bienvenida. = I feel welcome.
• Te sientes bienvenido / bienvenida. = You feel welcome.
• Nos sentimos bienvenidos / bienvenidas. = We feel welcome.
• Se sienten bienvenidos / bienvenidas. = They feel welcome.

In many parts of Latin America with voseo:

• Vos te sentís bienvenido / bienvenida.

Because mi here is the possessive word meaning my.

Spanish also has mí with an accent, but that is a different word meaning me after a preposition:

• mi vecino = my neighbor
• para mí = for me

So in this sentence, mi has no accent because it means my.