Apparently it means "bad/unlucky day", but several dictionaries I've consulted (both J->E and J->J) also say that it's also a Taoist term, as well as referring to a farmer's bad luck with weather. Is this a commonly used word at all? Or is it more specialized? Or none of the above?
- I can hardly imagine a native speaker who doesn't understand the word 厄日. I think average Japanese people encounter 厄日 once a year or so. Obviously it's not a common word on newspapers, but if you like reading novels, you may see this word more often.
- Today, 厄日 has very little to do with some specific religion, belief or tradition. It just means one's unlucky day that comes randomly. Its typical usage today is like this. This word is also used to half-jokingly comfort yourself or someone who encountered an unlucky event. "Don't worry, it was just your 厄日, you didn't deserve it." "Oh no, what the hell is this? Maybe it's my 厄日 today?" In this sense, I don't think this is an old-fashioned word.
- 厄年 looks quite similar, but it's a fairly religious concept to me. A large shrine often has a signboard that tells your 厄年, and they offer some 厄払い services/goods.
- You may probably say 仏滅/赤口 in 六曜 are also 厄日, but 厄日 is a more unspecific term. All the examples of 厄日 found on BCCWJ are clearly unrelated to the concept 六曜.
Yes it's a commonly used word (slang) for "unlucky day".
And yes, as you said, it originates from special term that means "day of ill omen (i.e. for activities)" in some schools of divination. This definition also survives today, but I can hardly imagine people use it in this sense except explicitly under such fields of topic.
I can partially answer your question by saying that this word is not common at all.
I just searched several corpora for the term, and it doesn't show up in the top 20,000 most frequent words in newspapers, nor does it show up anywhere in the top 45,000 most frequent words used on the internet.
I don't know anything about the word, but I can tell you that it's seldom seen in commonplace language-usage situations.