Regarding the first question, let me start by pointing out (as we have seen in a recent thread) that "to cast" also refers to "attempt to cast."
Here is the relevant part of the spell description:
BRB p. 495: The target unit must pass a Strength test every time it moves (included moving as a charger, fleeing, pursuing, moving compulsorily, etc.) or shoots or casts spells until the start of the caster's next Magic phase. If the test is passed, the target acts normally. If the test is failed, the unit is unable to perform the desired action, remaining in place and taking D6 Strength 4 hits from the net's barbs of light.
1. Since it specifies: "every time it...casts spells," the test needs to be taken every time the mage tries to cast a spell.
2. If the test is failed, the mage is unable to perform the desired action, i.e. he cannot attempt to cast this spell. However, he does not suffer from Broken Concentration, and thus can attempt to cast other spells, but not the one he tried to cast.
BRB p. 32: If a Wizard fails to equal a spell's casting value, or the total of the natural dice scores is 1 or 2, not only is the spell not cast, but the Wizard breaks his concentration and is unable to cast spells for the rest of the Magic phase.
BRB p. 31: Each Wizard can only attempt to cast each spell once per turn.
3. Extrapolating from the FAQ below, it would be the character that takes the test and the hits.
BRB Official Update Version 1.9, p. 14:
Q: If a character tries to leave a unit that is affected by the Net of Amyntok (or another similar spell or ability such as a Fulminating Flame Cage) including if he tries to charge out of it, is he required to take a Strength test? Also, if the test is failed, will any resulting damage hit only the character? (Reference)
A: Yes to both question. Though it is worth noting that a test is not required if he leaves without moving, through the Smoke and Mirrors lore attribute for example.
