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Re: Will other Lee Metford parts work on this rifle?
Yes, as will Long Lee parts.

Re: I have located one that could be razed for parts including 10shot mag, sights, bolt with dust cover and a decent stock.

You were asking about a dust cover. From your photo, your bolt already has a dust cover. You do, however, need a magazine cut-off.
For sights, I presume you are referring to a cape sight (ie 1 standing leaf, 3 folding leaves & 1 flip-up tangent to 1,000 yds.?

Re:What is the relationship between Andrews and BSA if Andrews didn't "sporterize" this sporter?

Andrews, like almost every other "gunmaker" bought in firearms from the real makers and had them marked, or marked them themselves with their name. B.S.A.Co. made 95% of the Lee Speeds, while L.S.A.Co.Ld. made the rest. This rifle started as a civilian "commercial" purchase, hence the lack of military markings. It was probably purchased for target use. At some time, the rifle was sporterized for hunting. The barrel was cut-down, a new front sight was made, the long range volley sight & boss were removed, the stock was shortened, refinished & chequered & the magazine cut-off was removed.

I've owned 3 Andrews & all but 1 were like yours; the remaining one was a #1 Pattern (engraved with a full-length rib)

Jim