Learn English – How to use must, May, Might for future and for present continuous

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!

Your initial examples are not as unambiguous as you think, and that is probably leading to later confusion. The continuous effect is not as baked in as you make it out to be.

"I'm having dinner at 2:00" actually sounds like dinner is scheduled to start at 2:00.

"I was having dinner at 2:00" could it was ongoing in the past at 2:00, or the 'was' could be subjunctive introducing doubt, meaning plans were changed ("I was having dinner at 2:00, but I missed the bus so we ate at 3:00"). Or even just that plans have been cast into doubt (eg "I was going to Toshi Station to pick up some power converters" -L. Skywalker)

"I will be having dinner at 2:00" could be grammatically be either ongoing or a scheduled start.

Your simple tense examples mostly work, although "I have my dinner at 2:00" sounds like it means "I have dinner at 2:00 every day" more normal usage for a specific upcoming event is "I will have dinner at 2:00"

Anyway, the main point is that the 'ongoing nature' isn't as firm a rule as you are looking for. So that's why there's nothing broken about "I'm having dinner at 2:00 tomorrow" with 2:00 being the start time.