I short
rog3/transmuter3/Arcane Trickster 4.
1. Will he qualify for forge ring ability? Technically he is 3rd level wizard specialist, and has 7th caster level. Why/why not. RAW suggest counting wizard, not caster level.
2. Familiar- if above character has a familiar, a pet stats are affected like 3rd or like 7th level/caster level of arcane trickster.
1) the prereq. is casterlevel 7 to you would qualify. As would a transmuter 5/rogue 2 with magical knack.
2. As I understand it the bonuses familiars get depend on the master's class levels, not the casterlevel. So add together all levels in classes that give you a familiar and you get the number for you familiar's abilities.
So unless the arcane trickster would give you a familiar, too it is level 3 for you.