"Yo miro a mi hogar" - would that be the same? Can I use "casa" and "hogar" interchangeably when I know it is my house/home?
I suppose you might not live in that house, you could be a landlord.
Edit: As Maxine374092 says below mirar means 'to look at' so you don't need the 'a'.
What is the difference between "veo" and "miro" for this sentence?
Normally, 'mirar' would require attention. In this sentence, they are full synonyms.
Sounds the same
From what you wrote, Santi_Minstrel, I'm supposing that you are saying that "seeing" is an autonomous process, while "looking at" is a conscious decision.
Why can't i write "Yo miro hacia mi casa"?
no need to insert an "a"..... e.g. veo hacia a mi casa