Is it obligatory to make up the prayers that follow the Maghrib and Fajr prayers, the validity of which is doubtful, or is each prayer independent?

If it is merely a doubt about passing wind, you are not required to make up the two prayers. However, if you are absolutely certain that they were invalidated, then it is obligatory to make them up only, and none of the prayers you performed in between them are invalidated. We advise against making up prayers unless there is absolute certainty of their invalidation, because giving in to obsessive whispers (waswas) leads to great evil, and the cure for them is to ignore them.