I recently wrote up Landmine Number, and I'd like to write more problems in the style of the story of this problem. Is it okay if I continue naming the sequels like LN II, LN III, LN IV? Side question, what is the appropriate amount of time I should wait until I put on the next problem?
Yes, there's no reason that wouldn't be allowed, as long as the problems are distinct enough that they're not duplicates. (It's ok if the theme is the same).
In terms of spacing - there are no rules governing this so it's up to you how you want to spread them out depending on your goals. I'd say wait at least a day though, to avoid swamping the new questions page, which could cause questions to get missed. I don't think there's any reason to have an upper limit though.