Your sons behaviour does sound quite age appropriate. It is common for kids to act out more with his parents. Ensure you keep the rules simple, clear and above all, consistent. You'll butt head to some extent all your life but consistency in love and discipline will keep the relationship good.

As far as daycare goes -- it sounds to me like you have a great balance, home with you part time and with other kids part time. Children need to socialize with other kids and daycare is a great place for them to do this. They also learn structure and routine which helps prepare them for school. Using drop-in centers are also a great way for social exposure that you can attend together and meet other parents also! Best wishes and congrats on the new baby!