1) Can supplemental actions follow the main action? Example: can I roll my Fists at -1 to attack, then move to an adjacent zone without rolling?
So long as the penalty is included in the full action, I know of nothing preventing it from being followed by a supplemental action.
2) Can counterspells and... let's call them spell-prolonging spells be rote spells? Rote counterspells probably don't make much sense, but I'm curious anyway.
My own personal opinions:
Counterspells I personally would flatly allow.
'Spell-prolonging spells' I would allow with the caveat that they would be specific to the prolongation of a single specific spell (though I would not necessarily define that spell to rote requirements).
3) Can sponsored Thaumaturgy spells (with Evocation's methods and speed) be rote spells?
This depends heavily on your group's definition of "evocation's methods". Such issues have previously been the topic of heated debate on these boards, from which a consensus was not achieved.
4) Do you consider acceptable making a stunt that gives a skill trapping a +3 while restricting its use? Example: would you allow a stunt that reads "you get a +3 to Craftsmanship when building weapons (or possibly even just melee weapons)"?
More restricted stunts yield greater bonuses.
Whether a particular circumstance is sufficiently restricting to warrant a particular bonus is often a matter of personal judgement.
For the example in question, I would allow the parenthetical version.