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!
As commented by Catija, the first three sentences are perfectly OK and as to the last one, the simple present tense is actually the correct usage.
According to grammar, when you are talking about the future, you should use the present simple in the time (when) clause, not a future form. For example, I'll come when I finish my work. So the correct sentence is:
I want to spend a year travelling when I finish university.
(Reference: The Free Dictionary - the use of "when").
Best Answer
In grammatical terms, both statements are correct. Which one you might prefer depends on the context and what you want to say or to imply. Take a typical situation in which a promising athlete is not training hard enough to win a race. Then you are more likely to remark to your friend:
The implication is that he does not want to put in the effort.
On the other hand, if you don't know whether the athlete is prepared to train hard, you might say:
This leaves open the question of whether the athlete does or does not want to.
Neither of these constructions refers to the past. They both refer to hypothetical situations in the future. But the first implies that the person concerned is unwilling to do what is necessary to achieve the goal although he/she has the potential; the second leaves it open.
An alternative is to say: He can do it if he wants to. It's a more direct way of speaking. He could do it.... is a more guarded or reserved way of saying the same thing. It implies that there's a condition to be met.