You must be rated and current on one of the ezy types (Airbus or B737) to be offered direct entry FO deal.. anything else and it's the TRSS on offer.
As regards the AHC, as far as I can make out from the recent groups going through selection, it either comes down to performance in the sim assessment or possibly your current status. For example of the 7 people I know who went through recently (4 captains and 3 F/Os) all were accepted for TRSS, only the 3 F/Os were required to go for the Advanced Handling Course. The F/Os were all medium jet rated with several thousand hours. One of the captains was a turboprop guy (but had done a 737 rating many moons ago, no time on type though).
Hope that's of some help.