Can you use "(sic)" in other contexts?

Generally "sic" should be in square brackets, not parenthesis, as an editor's mark. Parenthesis are acceptable, but because square brackets are less often used, the intent is more clearly conveyed. Also, sic should not be italicized nor should a period be placed after it as it is not an abbreviation. (Reference)

Sic is derived from a Latin adverb sīc meaning "so," "thus," or "in such a manner."

I personally would not use sic to indicate unusual or ironic meaning. In the case of the pilot,

The flight was canceled because the pilot forgot his lucky dime!

The exclamation point conveys the meaning clearly. It's an outcry. Compare to:

The flight was canceled because the pilot did not have her security credentials.

The presence of an exclamation point on a statement conveying a usual situation would feel out of place. However, if this were being spoken by someone who was enraged by the canceled flight, the exclamation point could be added to convey emotion and frustration.

In the above cases, punctuation serves to convey the intended meaning by context alone.

That would be an ironic extension of the regular usage of sic. I'd interpret it as "yes, dear reader, surprising as it may be, this is what he did". I have seen it used like this on occasion. It looks a bit silly; I'd never use it myself. There exist several little jokes that are made again and again by journalists because they feel they need to decorate their texts; I'd skip them, unless they are new or actually funny. But perhaps that is just grumpy old Cerberus whining.

I have not answered before, English not being my mother tongue (while in Italian, my language, the usage of "(sic)" described by the OP is quite frequent, to the point that some people don't know its original use, and confuse sic with sigh, as if the author were sighing for the blunder or funny episode he is telling).
That said, I have just found in a text by Damon Knight, who knew his English, such a use. In an article later collected in In Search of Wonder he berates several aspects of A.E. Van Vogt's style, plotting, language etc. In particular, he is describing some incongruences in Van Vogt's The World of Null-A. In one of these, a gang hostile to the main character is temporarily helping him all the while, Knight writes,

keeping him under control by means of a “vibrator” which changes the atomic structure of everything around him so he cannot memorize it (sic) and effect an escape.

So, apparently, Knight's use of "(sic)" is exactly to stress the absurdity of the situation he is relating.

Based on my experience, using sic to draw attention to unusual characteristics of a sentence would be considered unconventional in English. Also highly unconventional would be the use of an exclamation point following sic within the brackets or parenthesis.