If it's a sentence like "I'll see you next week." or "What are you doing next week?" Then "the next week" would be wrong. If it was a sentence like, "We expect a hurricane to arrive sometime in the next week." then "the next week would be correct. Basically, you only say "the next week" if "IN", "DURING", or "FOR" is before it. I'll be here for the next week. ["I'll be here all week."] "It will happen sometime during the next week"

I think" next week" is based on the present point in time , while "the next week" is based on the past or future point in time.

It depends on the situation. Por ejemplo, "I will see you next week." = Te veré la próxima semana. "I am not getting paid this week, but the next week." = No me pagarán esta semana, sino la próxima. There are situations where one version or the other may be correct. Good luck!