Your final schedule above looks good. I think it's doable with a child. Good luck.

One more thing..the on again off again bus in Barcelona is great (be sure to stay in town, takes too long even by subway to get into town)! Specially with a child to get a view of the whole city. Not so much the one in Madrid.

At the moment the barcelona hotel I booked is near sants or maybe few blocks away. But it is cancellable. Any recommendations? the other hotels are either too expensive or below 3 stars which doesn't look decent.
Apartment is also ok if clean and modern.

you'll have to take public transit all the time from Sants

Try habitat apartments
www.habitatapartments.com

Yes Santsstation is not a centrallly located station but on the edge of Madrid- not really dreamy area (not bad) but stay more near Atocha station for convenience to rail station and be in the heart of things.

Sants is in Barcelona not Madrid.

I also warmly recommend the hop on and off bus (If I remember, there are 3 lines and you can switch between them). The sites in Barcelona are really spread out and it makes it easy to get around. It also helps plan your visit.

Sants is in Barcelona not Madrid.>

Yes to me similar stations -stupidly mixed up stations -thanks for correcto.

That should have been Sants I find similar to Madrid Charmartin.