What is the ruling on sitting with people who commit sins with the intention of inviting them to righteousness, and what is the ruling on sitting with them for a period without inviting them, in order to strengthen the relationship before inviting them?

Sitting with disobedient people is not without two scenarios: The first is sitting while they are committing a sin, and this is impermissible unless you are able to invite them to righteousness and denounce their actions. The second is sitting with them when they are not committing a sin, and this is permissible, God willing. As for sitting with them for a month without inviting them to righteousness, it is permissible as long as it does not involve condoning the sin, and this might be an entryway for them to accept your advice later.