Is there anything to show if Shmi Skywalker (Anakin's Mother) was Force-sensitive?
And, along that line, is there anything to indicate that the whole "virgin birth through midi-chlorian fiat" or the "virgin birth via Plagueis playing with the Force" would not be possible unless she were Force-sensitive?
It is probably safe to say that she wasn't Force-sensitive, mainly because it further adds to the legend of Anakin, but I don't think it is ever truly established.
As far as your second question pertaining to the possibility of a non-Force-sensitive producing a messianic Force birth, The Darth Plagueis novel makes two things clear relating to midi-chlorians:
Midi-chlorians exist in all lifeforms, this is shown when Plagueis manipulates King Veruna's midi-chlorians (who did not possess enough of them to be considered to be Force-sensitive), which in turn slowly killed him. The other item that the novel establishes, is that Anakin was born as a direct result of midi-chlorians thwarting Plagueis' attempt at tipping the balance of the force towards the dark side.
According to the Star Wars Episode I: Visual Dictionary
Shmi can always sense when Anakin is nearby, even when she cannot see or hear him
I think Shmi Skywalker is Force sensitive, just like Chirrut Îmwe
Per the Legends* factbook Star Wars: Graphics, Shmi is not a 'Force Sensitive' (e.g. someone who can actively sense the Force) as she possesses a midi-chlorian count of just 3,300, less than half that needed to be classed as such. That being said, her slightly elevated numbers do suggest that she has a more general sensitivity to the Force.
*Astoundingly, these figures seem to have been taken from the fake midi-chlorian numbers invented by noted Star Wars fabulist SuperShadow and should be treated with great caution.
I think she is. When Anakin proposes he race for Qui Gon she says that Anakin is meant to help Qui Gon. I just can't think of any reason George Lucas used that wording given we know he changed McGregor's dialogue.
