You would want to get a finch cage with no more than 1/4" bar spacing.
As for letting the little one fly free all day that might not be the best thing especially if you have other birds or pets.
You would have to make sure that windows are covered and that she/he understands windows and walls so that those
won't be hit and she/he won't be injured. Bathroom doors would always need to be closed and if you are cooking in
the kitchen best to keep the little one in the cage. I would advise free flying indoors only if there was supervision and/
or monitoring. I would still wait a bit before moving the little one into a cage.
That "cage" in the photo is super small and way too small, IMHO. Spee started out in a paper towel lined bowl and was then moved to a 5 gallon
terrarium that was lined with paper towels with heating pad under half the terrarium. The paper towel lined bowl went into the terrarium, too,
and he was placed in that but could jump out if he wanted to. That little bowl stayed with him even when he was moved into a cage.
I don't remember how old he was before I moved him into the cage but I do believe he was at least 4 weeks
old or a bit more. Spee had issues with his legs and I was very careful moving him into the cage. I would have him in there a few hours when I could
monitor and then put him back in the terrarium. At one point I was very busy and I think two days went by and I didn't try him out in the cage.
He had a huge temper tantrum, banging on the glass and screaming so I moved him into the cage for a few hours. When I put him back in the terrarium
the same thing happened and I knew, at that point, he was done with the terrarium. In his own way he let me know he was ready to stay in that cage and not
be returned to the terrarium.
She/he could be chirping because she is still hungry or just wants more attention.