Some thoughts ..
(a) AEO ... the aim is to fly the aircraft so that the flight path stays above the calculated OEI flight path so that there is no particular concern in the normal takeoff case .. however, one needs to watch IAS critically if there are turns in the procedure as the turn radius varies with speed.
(b) while AEO routine cleanup procedures vary, typical protocols see a gentle acceleration (ie reduced ROC) from heights varying from 800 ft. Generally with twin jets, the AEO performance is such that, by the time the initial climb is established etc., the aircraft is going to be at a reasonable height arp anyway ... we all understand that this doesn't apply to some of the wonderful old girls .. Fokker, Argosy, 748 etc. etc ...
(c) minimum (OEI) third segment (acceleration) is 400ft arp
(d) maximum third segment is only limited by the AFM and commonsense .. so, for instance, Dart powered machines I have played with have a 600 ft maximum due to feather pump limits. Many jets can be pushed up to 800ft or more as necessary and considered appropriate. The general limit is that height which results in the END of the third segment occurring within the time limit for engine operation at takeoff thrust (generally either 5 or 10 minutes)
(e) often an operator will adopt a standard height based on a critical runway for standardisation (the Australian domestic airlines did this years ago for the jets using 800ft based on Canberra departures which were critical for the DC9.
(f) your particular operator's SOP for a given runway should prescribe the third segment height, either specifically or as a generic level for that operation ... he who doesn't follow that procedure exposes the aircraft to potential problems .. obstacle clearance in particular .. the pilot generally has NO idea what sums went into the procedure or what obstacle(s) are critical and under what circumstances .. be aware that the critical obstacle may well change depending on OAT or W/V