This is why the linguists insist that English has two tenses: past and non-past!
These uses of what we ordinarily call “present” tense, simple or progressive, with future reference (instead of the explicitly futurive will) tend to be restricted to definite plans. They say in effect “This is what is on my schedule”.
Q: Sherry, is Bob free sometime tomorrow for a quick review?
A: Let me look at the book ... He’s in meetings til noon, and he’s out for lunch, but he can see you at three?
So sentences 1 and 2 are both acceptable, and there's no real difference between them. Discourse context will contribute to determining which you use (or whether you use will), but there's no rule you must follow.
Similarly, your final example, ‘But what do you do next year?’ I asked. ‘Yes. That is the problem,’ he replied is pretty ordinary. Questions of this sort arise, typically, when someone has described an action to be taken in the present or near future and you want to know what action will be taken in the longer term to account for the first action’s consequences. In effect, it asks “Do you have a plan for next year?”
The two questions, 3 and 4, are a bit different, because the phrase every day establishes a different sort of context for use of “present” forms. As you know, these forms are ordinarily used to describe habitual and repeated actions; every day reinforces that interpretation, and collides with a futurive reference. So these sentences are very unnatural. You might just get away with 3, Do you know what you are doing every day this summer?, if you are trying to find out if your interlocutor's calendar is fully booked. But I cannot imagine a context in which 4 would be natural; it suits better with a present referenc, something like this:
Do you know what you do every day? You leave the cap off the toothpaste every goddamn day!
CGEL claims that
the non-progressive I phone her tonight “suggests a schedule or plan” and therefore would not be employed (“it’s hardly possible”) if the speaker had only a casual intention of calling: an intention of the sort expressed by I’ll phone you tonight.
The progressive I’m phoning her tonight can, like the non-progressive be employed in the case of a schedule or plan; but unlike the non-progressive it can also be used of a casual intention.
I agree with CGEL, and in fact would go farther; the non-progressive does not merely “suggest a schedule or plan”, it is (in my US experience) only used of a single event when a schedule or plan is the topic.
According to my calendar she sees Bob in New York this afternoon ... I phone her tonight ... depending on what we sort out you and Carol work up the numbers overnight and email us ... then we've all got a conference call at 8:00 in the morning. 8:00 our time.
The reason, I suspect, is that a very strong non-habitual context, shared by both parties to the discourse, is required to overcome the ordinary habitual/generic implicature of the non-progressive present construction.
Best Answer
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.