Is it considered a sin to take from a sum of money given to purchase a gold gift for a newborn, and use part of it for household expenses or newborn necessities?

If your father gives you money to buy gold for the newborn, the preponderant opinion is that it is not permissible to spend it on anything other than gold, unless with his permission. However, if he did not specifically intend for it to be spent on gold, then there is no harm in spending it on other things. The donor's intention can be understood from the surrounding circumstances or by asking him.