Depends if the parents are WNA/WNA, WNI/WNA or WNI/WNI.
You have the Akta Kelahiran (akta lahir) and obviously the child needs a KITAS if both of you are foreigners. From the title's mixed marriage I understand it's a WNI/WNA: In that case, you'd need the Affidavit for the foreign passport (very advisable to get one).
NB: with the current situation it's not so easy to get passport etc. Better to wait till after the 2nd week of April. And even then it might be a long process...
Well if its open it is common courtesy to attempt to inform them, but probably they are not open until April 5th anyway, but yes of course inform them at a suitable time, unless the new born needs a passport so soon which is unlikely
For WNI/WNA marriage, if you do NOT arrange for WNA passport for the child, you don't need to report to Imigrasi right away. I didn't report my last two until I made WNI passports for them 6-12 months after they were born without any issues.
If you do arrange for WNA passport, then you need to get affidavit right away.