1. It's up to you whether or not it's handwaved! You can say that the character woke up on their own or include their being unconscious and lying in the middle of the road as a prompt for others -- whatever floats your boat. But characters are pulled exactly as they are based on canon point, so they would be unconscious if that was indeed the chosen canon point.
2. It depends on the source of this power. In most cases it is limited to high-frequency worlds because it's not based on biology or factors into survival, but a robot character that had a hammerspace-type function built into their frame may be able to keep the power. It really does depend on the situation the character is in specifically.
3. The network cannot be turned off completely. Once characters master the telepathy they've been given, they won't accidentally broadcast and will only use telepathy purposefully (this varies from character to character and their magic aptitude), and other characters with great magical aptitude can help to sort out the voices in their head more easily, but characters will always have access to telepathy (both sending and receiving) unless an event or world makes it impossible somehow.
no subject
2. It depends on the source of this power. In most cases it is limited to high-frequency worlds because it's not based on biology or factors into survival, but a robot character that had a hammerspace-type function built into their frame may be able to keep the power. It really does depend on the situation the character is in specifically.
3. The network cannot be turned off completely. Once characters master the telepathy they've been given, they won't accidentally broadcast and will only use telepathy purposefully (this varies from character to character and their magic aptitude), and other characters with great magical aptitude can help to sort out the voices in their head more easily, but characters will always have access to telepathy (both sending and receiving) unless an event or world makes it impossible somehow.