Is it permissible to spend hours on the internet in permissible activities for someone who has to make up prayers that were not performed correctly, and does the permissible become forbidden if it distracts from an obligation?

It is not permissible for one who has missed prayers to engage in permissible activities that prevent him from making them up, and for which there is no need. If his engagement in supererogatory prayers is prohibited, then permissible activities are prohibited a fortiori.

He must hasten to make up what he owes, dedicating his time to that, as long as it does not harm his body or livelihood.

Indeed, engaging in permissible activities becomes unlawful if it distracts one from an obligatory duty.

Making up prayers where a person, out of ignorance, omitted some of their pillars, is a matter of scholarly disagreement regarding its obligation, but making them up is more cautious.

As for one afflicted with obsessive whispers (waswas), there is no blame upon him for acting upon the easier opinion.