To answer your first question,
What do we know about Shirou's biological parents/grandparents/etc.? Do we even know anything at all?
I would like to say that there has been no information released regarding it in the anime/manga/visual novel/games/wikia site. However, there is this piece of information on the wikia site.
Ten years before the start of Fate/stay night, Shirou was an ordinary boy living with his parents in Shinto. The great fire caused by the contents of the Holy Grail spilling out at the end of the Fourth Holy Grail War killed both his parents and left him mortally wounded within the blaze.
Therefore, we do know that his biological parents died during the Fourth Holy Grail War.
To answer your second question,
Understanding why Shirou can perform magecraft despite presumably having had non-mage parents.
Again, the wikia site states that
Shirou persistently asked Kiritsugu to teach him Magecraft. Kiritsugu eventually complied even though he disapproved of Shirou's desire to learn his craft.
That is why Shirou Emiya can perform magecraft.
To answer your third question,
He seems to never say anything about his parents.
All I can say that there is no definitive answer to this question owing to the fact that there is no reference to this in any of the aforementioned media.