La música calma a mi gato en casa.

Spanish requires the personal a when the direct object is a specific person or beloved animal. Since mi gato is a definite, animate object, you must insert a: calma a mi gato.

No. Omitting the personal a with a specific pet sounds ungrammatical in Spanish. The a signals that mi gato is a living being.

Calma is the third-person singular present indicative form of the regular –ar verb calmar.
• calmar = to calm/soothe
• yo calmo, tú calmas, él/ella calma, nosotros calmamos, etc.

Spanish often uses the definite article before abstract or general nouns.
• La música (music in general or a known piece)
• Without an article, música alone can sound like a heading or title rather than part of a sentence.

En casa means at home (in one’s own home). When referring to one’s usual residence, Spanish drops the article:
• Estoy en casa = I’m at home
If you say en la casa, it usually specifies a particular house, not necessarily yours:
• En la casa de María = in María’s house

Yes. You can use the direct-object pronoun lo (for a masculine noun) and say La música lo calma. You can even combine both for emphasis: La música lo calma a mi gato.

You can, but it shifts the nuance.
• La música calma a mi gato → a general or habitual statement (“Music calms my cat”).
• La música está calmando a mi gato → focuses on the action happening right now (“Music is calming my cat [at this moment]”).