I get the sense that "agarrar" is more to hold as in (hold on to, don't drop the baby) and "sostener" is just hold  (in your arms/hands)

In the sentence Cámbiale el pañal..  the Pañal is the direct object (it is being changed), and I am pretty sure that the you are correct that the baby is the indirect object.   In there conversion the baby might have been mentioned earlier and we only see this one sustenance or it is assumed, I do not know. But I am sure that the sentence can be literally translated as :

To (for) the baby change the diaper.

 Un amigo acaba de eseñarme cómo poner una photo en esta seccíon. Esta es el bebé de un amigo mío que se acaba de nacido recientemente. En esta foto el bebé tiene sólo pocas horas de edad. ¡Mira, este bebé nació con una sonrisa en su cara! 

A friend just showed me how to post a photo in this section. This is the baby of a friend of mine that was just born recently. In this photo the baby is only a couple of hours old.  Look, this baby was born with a smile on his face! 

Una pregunta: puede decir "acaba de nacer recientemente" was just born recently

o es mejor decir: nacío recientemente?

You can say "que acaba de nacer" or "que ha nacido recientemente".

Acabar + de + infinitivo= This structure already implies that it just happened, so you don't need to use recientemente.

Muchas gracias!