Learn English – Present simple for expressing the future

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!

Addressing your specific questions:

'Can you imagine how dirty it gets over time?'

When people talk informally, they are often a little sloppy. The present tense would be the perfect choice to state a generalization or natural law such as:

'White gets dirty over time.'

This idea was close enough for the speaker. Someone who likes to be more precise might say, instead, "Can you imagine how dirty it would get over time [if we bought the white pillow]?" But you might not enjoy living with someone who uses English precisely all the time. Such people tend to be nitpicky.

'The grease from your fingers will burn into the bulb and then it breaks.'

Here, two ideas have been spliced together into one sentence.

First idea: "[If you touch the bulb with your fingers] the grease from your fingers will burn into the bulb."

Second idea: "When that happens, i.e. when you touch the bulb with your fingers, the bulb breaks."

This is another example of the simple present being used for a generalization or natural law. And again, in a situation of informal speech you caught the speaker being a little sloppy.

'I hope it goes away overnight.'

I don't know how the English grammar experts would view this (if you want to know, you could ask over at ELU SE), but I will share how I see this example. My other primary language is Spanish, which has a subjunctive. English has one too, but people don't think about it much. In Spanish you really can't get away without thinking about it. In Spanish, this would be

Espero que se quite para mañana. ("se quite" is conjugated in the subjunctive)

I see the English sentence the same way -- I see goes away as the present subjunctive, which happens to be conjugated the same as the simple present.

Your questions were good, and you've done some careful listening and recording of what you've heard.

I would be remiss if I didn't point out a small but important misuse of the simple present tense in your question:

You wrote: I moved from Germany to California and since I'm here I hear people use the simple present....

Expressions beginning with "since" are notoriously easy to get tripped up on.

Better: I moved from Germany to California and since I've been here I've heard people use the simple present....

Or: I'm from Germany. Here in California I hear people use the simple present....