Learn English – How to write a date at the beginning of a sentence

Your instincts are exactly right. English does not have a true future tense, and there are many cases where a past-time statement requires the past tense but an analogous future-time statement uses the present tense. For your examples, I would write:

1. Today is 4/17. 4/7 was 10 days ago.
2. Today is 4/17. 4/27 is 10 days from now.
3. The deadline was 4/7, which was 10 days ago.
4. The deadline is 4/27, which is 10 days from now.

(Note: I've changed ahead to from now, since the latter sounds more natural to me. Ahead is not wrong, but it's a bit literary, and it clashes a bit with the bureaucratic and prosaic context here.)

[…] however, for #4 using 'will be' suggests the deadline is not yet finalized at the time the statement was made.

Right — or perhaps that the deadline has been finalized, but has not yet been announced, or is only now being announced. It's not a normal way to refer to a previously-announced deadline.

In American English, we don't normally write dates that way, but instead write and say with the month first — even if it's written day first, we'd still usually pronounce it month first. So "27 July" becomes "July twenty-seventh".

I believe other dialects usually, but not always, include the article before the day when putting the day before the month, so "[the] twenty-seventh [of] July". It will be understandable no matter which way you say it, but it's safer to use the slightly longer form to avoid sounding awkward in places that aren't that terse.