Re: Aqeeqah party
Well it depends - DO you want more gifts 
You could always state on the invitations that there are no gifts being accepted, and to make a donation to charity in your childs name instead. Lots of islamic charities do islamic charity gifts and you can get a gift certificate for example.
As for age of newborn, I guess it depends on how well you’ve recovered. My baby will be under two months by the time of the party, I would’ve had it sooner but he was born during ramadan and by the time I’d gotten round to organising it after eid etc etc. Obviously you want to wait long enough for you (and baby!) to be well but not so long after birth that people have already been to visit you as I think that is when people start assuming that you are after extra presents - they bring a present when they visit you at home and obviously social decorum dictates that they shouldn’t come to the party empty handed either.