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How should I make an unconctracted (negative) question with cannot: Cannot I do? or Can I not do? (if there was other aux instead of "can", the 2. would be correct; but cannot is written together; is it true in negative question too?).

Formally: Can I not do?
Informally: Can't I do?

'Cannot I do?' is archaic

Does it mean generally speaking that when I have to split "cannot" into aux + subject + not, I should always write "can ... not" (write cannot as two words, so a formal question tag: "I can do, can I not" and adverb: "I can <some adv> not")? (Am I right? Or should be "I can do, cannot I?")

I'm not sure that I fully understand your question, but let me just reiterate that the sequence 'cannot I' never occurs in contemporary English. The choice of interrogative forms is always between the two previously mentioned.

Does that answer your question?

I think that:
1. A formal tag question to can should be "I can do, can I not?" (but not "I can do, cannot I?")
2. If I want to put an adverb beetwen cannot I must write it separately: I can <adv> not do. (but not I cannot <adv> do)

Am I right?

I think that:
1. A formal tag question to can should be "I can do, can I not?" (but not "I can do, cannot I?")
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Yes, although you would need to say "I can do IT, can I not?"
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2. If I want to put an adverb beetwen cannot I must write it separately: I can <adv> not do. (but not I cannot <adv> do)

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No, the adverb would be placed before or after 'cannot', as in
I really cannot say. or I cannot absolutely guarantee it..

The construction [can + ADV + not] is rarely used.

One more question: what's the general rule - when to write can + sth. + not, and when cannot + sth.?

There is only one situation that I can think of where you would have to use [can+ADV+not] rather than more standard [cannot＋ADV], and that is in a sentence such as

If you're worried about the results, you CAN always NOT go for the medical in the first place!

meaning 'it is at all times possible for you not to go', clearly distinct from near-meaningless

?If you're worried about the results, you CANNOT always go for the medical in the first place.

where the second part would mean 'it is possible that you may not be able to go at certain times'.

Other than this fairly rare case, however, [cannot + ADV] will invariably be the correct/preferable form.

Thanks for Your help, Alan. It is very appreciated.