Yes, (1) is a short form of (2). The subject I and the auxiliary verb will are both understood to be elided. (Otherwise it wouldn't be grammatical: a future tense needs will and sentences need subjects.)
"I will be seeing you next week" is almost the same; however, there is slightly greater feeling of certainty. Your doctor might say that if you have a follow-up appointment in one week's time to emphasize that, or that same person could just use "see you next week".
By contrast, it would be somewhat unusual to say "I will be seeing you next week" to coworkers on Friday evening, whereas "see you next week" is fine. The reason is that we don't have a specific appointment to meet coworkers next week; it just happens because we show up for work. It's simply too many words to use for something that is passively expected to occur.