Is it permissible to postpone the 'Aqiqah for a newborn for several months, or is it obligatory immediately after birth, and is it necessary for the father to appoint someone to perform it on his behalf?

The aqiqah is a recommended Sunnah, not an obligation. The Sunnah is for it to be on the seventh day. There is no sin upon one who delays it or abandons it. It is permissible to delay it beyond the seventh day, and there is no sin in doing so. It is preferable to offer it if possible. There is no objection to delaying it due to an excuse or to achieve a more complete benefit. If it is not possible to slaughter it at its prescribed time, there is no harm in delaying it to another time as circumstances permit. If it is possible to appoint someone to slaughter the aqiqah at its proper time, that is more appropriate and better. If that is difficult, there is no harm in delaying it.