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!
I'm not sure if OP's second "single sentence" version is technically a zeugma/syllepsis, but it's in that general area (the "deleted" main verb after have is kept, which doesn't match the "retained" form keep).
In practice native speakers do sometimes use such forms, but as a rule they're avoided. OP should use the first version, explicitly retaining the two different (past and present/future) verb forms.
Probably for purely stylistic reasons even OP's first version sounds just a little strange. Most native speakers would probably prefer to separate the two tenses even more explicitly...
I have kept this thing between us and will continue to do so
To illustrate exactly why the construction is problematic, consider...
"I have and will give you money"
...which could be interpreted as "I have given you money [in the past] and will do so again [in the future]" or "I have money, which I will give to you" (perhaps I never had money before, or never gave any to you).
Best Answer
Using the form with understand emphasises that I understand it right now. The form with have understood merely states the fact that two events took place in the past. Of course in the normal state of affairs if you understood it in the past you still understand it now but sometime people forget.
Either is correct thought and effectively mean the same just the emphasis is slightly different.