Since we have a single word like birthday, is there a corresponding term for the date of someone’s death?
I think the closest word is obit, which the Oxford English Dictionary defines as "A record or notice of a person's death, or of the date of it; (also, occas.) the date itself" (sense 1(b)), and further as "An office or service, usually a mass, held to pray for the soul of or otherwise commemorate a deceased person ... on the anniversary of his or her death, or at some other appointed time; a yearly or other regular memorial service" (sense 2(a)).
The second definition, referring to a yearly commemoration, shows that "obit" is a reasonably good analogue for death of "birthday" for birth.
EDIT: The OED also gives obit day, which might be a closer analogue (yes, it's not a single word...). An example quotation from the OED, dated 1995, is: "The communar account rolls record that £1 was spent on his obit day between 1347 and 1536." However, if you allow two-word analogues, the OED also defines death day as "The anniversary of the day or date on which a person died" (sense 2).
Well Birthday and Deathday. It's not an official word. But I don't see why not. You also say deathbed don't you? Yet I have never heard of birthbed. Though it is the same thing. Only you don't ever actually lie in your birthbed, your mother does, but she is not born in this bed. With deathbed you are the one lying there so it is more frequently used.
In a word, no.
If we commemorated deaths by a regular annual remembrance then there might be such a word. We only have the word birthday because we do commemorate births. But the day we are born is not our birthday! It is our date of birth, just like the other end of things is called our date of death. Our first birthday occurs one year after our date of birth.
German has two single words: Geburtsdatum and Sterbedatum. But then, German is possibly the most agglutinative language on the planet.
Completely off-topic now...
My favorite long German word? Donaudampfschifffahrtsgesellschaftskapitän.
Means a "Danube steamship company captain." This is not, however the longest German word. Some candidates are discussed here.