How was Gamera captured by the Tangee? How long has it been? How does Char feel about rescuing her? What would he give up to do it?
Tangee used the simplest method to draw her attention - a living bait, another prisoner of their, villain named Buster. As you can guess from its villain name , he gained enough trouble to draw attention of Gamera, and later she was captured by their ambush. It was around three years ago. Since Gamera had sort of backup plan, Lev had more than enough money to go to literally any school he wanted and buy everything he wanted, however he was rather conservative with her funds and was spending the money accordingly.
Char from one hand want to rescue her, but on the other hand he know that she probably wont be very happy about the fact that he sticked to the team, so he will carefully ask others, explaining the situation and possible workarounds before saving her. He is not afraid of her, rather afraid of the fact that he is not a match for her power, even considering the fact that Gamera uses older games(around a 20-30 years older) than Char, usually recognizable by clearly visible pixel particles(Char games is closer to reality, if this term could be applicable to a videogame stuff). Char believe that its due to some sentimental value rather than inability to learn them. So probably he will look other ways to convince her, however he will still protect other teammates from her if the peaceful solution is impossible.
Its better to say what he will not give up. Thats his team, his friend Billy and innocents. Other than that, he is ready to pay the price, believing that he will manage to reason Gamera. And maybe sometimes they will became just a mother and son, like Lev always wanted to.
How does Char best live up to his mentor's expectations? In what way does he disappoint her the most? What kind of hero does she want him to be?
Before she went missing, Gamera was very proud of him just because he trusted her, even despite she was literally torturing him with various exercises that could break body and mind. She is still believes that he is perfect vigilante that doesnt require any teammates, so she will ensure that she show Char how miserable and rule-breaking they are. She will make them unworthy of Char and his destiny. Together with all this she will give reasonable advices about what the particular person did wrong, how to fix that and etc., so technically she may change her mind if team manage to pass her trials. She might be a good mentor if she didnt had such awful personality.
The biggest disappointment of Gamera in Char is the urge to protect others. Regardless of challenges she gave to Char, he havent gave up this disadvantage. However, Gamera thinks that either she will manage to fix it, or Char will face a grim reality once or twice and give up. Gamera see relationship with others as a way to affect herself, so she thinks about friendly connections as weakness. Char thinks that she may hide some trauma acting like this.
Gamera see Char as vigilante that one day probably surpass her in her job, so she still have a lot of expectations towards him. If he gave up his friendliness and desire to protect others, he wouldve been perfect protege, yet she dont want to force or brainwash him in any way, believing that he had to arrive to this decision by himself. But sometimes she doesnt mind to show him the right direction.
How did Char keep in touch with Billy? Why has their friendship endured for so long? What kind of hero does Billy want him to be?
Since Lev and Billy is almost as brothers, they are very close friends. Lev was often protected by Billy when he was in orphanage, so Lev feel a mix of appreciation and gratitude towards him. Also Billy is a valuable piece that didnt let Lev fall apart under hard trainings of her mentor. Shortly after he gained some pocket cash from Gamera, he immediately bought his friend a phone, and they often having chat through either call or video if they just cant go somewhere together.
Billy wants Lev to be strong enough to protect others, without giving up on everyone, so despite Lev wants to meet Gamera's expectation, he also want to meat Billy's expectations as well. Also Billy believes that Lev might be a good leader, but Lev isnt a huge fan of this idea, because he thinks that he doesnt deserve to be the leader.