Must the prayers of the past four months be re-offered if one suspected something had exited him, then it later became clear, after some time had passed since the repeated incident, that a drop of semen had exited, knowing that most of the prayers were performed in congregation and that the fluid did not yet reach puberty?

Doubt regarding the emission of semen does not necessitate ghusl (ritual bath). The default state is not being in a state of janabah (major ritual impurity) until certainty is achieved. Therefore, there is no obligation to repeat the prayers of the past four months. The emission of semen, even a single drop, is a sign of puberty and necessitates ghusl. As for doubt concerning puberty, it is not legally established due to the lack of certainty. It is obligatory to lower the gaze from what is unlawful to look at.