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!
Both versions are grammatically correct.
It is possible (although in my experience relatively uncommon) to use "will" (or "would" for past tense) to indicate habit, but only with action (i.e. non-stative) verbs. This construction is probably more common in the past tense. When talking about the present, the form with the present simple seems more natural to me as a native speaker and is more concise, but the meaning is exactly the same either way. So you could say:
Every day after school Jimmy goes to the shop to buy sweets. OR Every day after school Jimmy will go to the shop to buy sweets.
or in the past tense:
Every day after school Jimmy went to the shop to buy sweets. OR Every day after school Jimmy would go to the shop to buy sweets.
However, in the following example with a stative verb, only the present simple/past simple are possible:
In high school I loved science. NOT In high school I would love science (ungrammatical)
Another way of expressing habit in the past tense is the construction "used to". I'm not going to convolute this answer by going into that more, but there's more information here and here.
Best Answer
There's been a significant usage shift in this area over the past century or so. Consider this NGram...
To the extent that there might be a shade of difference between including the explicit "future" indicator will or not, I'd say we're perhaps slightly more likely to include it when there's some "distance" between the speaker expressing his hope, and the (later) time when the addressee has a (positive or negative) reaction. So you might think including will is more suitable if you're writing a note accompanying a gift being sent through the post, for example. But most people wouldn't think of or notice such fine nuances.
Note that this issue only applies to a few verbs (hope being one of them). With other verbs, such as I know / expect / think / etc. [that] you will get the job, the word will must always be present.