Some clarifications please!

All of this should be covered in the in-game manual ... but to clarify ..

Set-piece specialist - is this freekicks, corners and penalties?

Yes.

Weaker foot - Do players have hidden preferred foot? Some PPMs reference 'weaker foot'

Yes.

Preferred Moves - How does this screen work? Are they all moves my player has? Some of mine seem to have 20 when you check each playing position. Why are some green and some red?

Players have a set number of preferred moves, some are only present if they are asked to play in a specific manner and might not be active for 'all' roles.

For instance a player might 'Run with ball' when asked to be a winger because he's intelligent and realizes it helps that role, but if asked to be a holding midfielder he might surpress that aspect of his character (some players however might not be able to avoid their natural characateristics and continue to run with the ball even if its inappropriate).

Items shown in green are considered 'good' for the currently selected role, those shown in 'red' are considered bad (ie. run with ball for a holding midfielder).

Training categories - Can we get a breakdown? The full game used to have some surprising ones. eg GKs needed high tactic training for positioning or something

I think they're listed in the manual - I'll check later if I remember to.

Team work attribute - Is this simply greediness? Is there an aspect of Work Rate, or is that missing completely from FMM?

Its a combination of greediness and work-rate in FMM.

Counter-attack - Does this include some aspect of soaking up pressure? Or is it just the aggressiveness your team has once the ball is turned over?

Counter attack also attempts to lure the opposition onto you and is mainly about hitting hard and fast when you get the ball and they're sucked forward ...