Andy's questions (renumbered where appropriate for ease of reference) with my responses in italics:
1. Did Stratocaster #34346 come with flat-wound strings?
Unknowable. Some USA players do insist that late 50s Fenders came with flats, but Hank has stated that he has never used them. On the other hand, there is good photographic evidence of flatwound strings on the Antoria in use during the first few months of 1959.
2. If so, was Apache recorded on flatwounds? Or well-used roundwounds?
Roundwounds. Hank isn't likely to have got that consistently wrong and he'd have changed the strings a good few times between mid-1959 and June 1960.
3. Did Hank continue to record on # 34346 after the Shadows received the rosewood-board Strats?
Unlikely. Guitars were just guitars in those days and it is likely that the newly-arrived rosewood board Fenders had slimmer, better feeling, necks and actions than the one on 34346. I know which one I would have used...
4. What was the last piece recorded with #34346?
Unknowable without knowing the date of arrival of the matching red Fenders. There was a lot of Shadows recording in the period May - June 1961. The Shads were using 34346, the sunburst Jazzmaster and the sunburst Precision Bass when they appeared on TV to promote F.B.I. but the all-red set when they appeared on "Crackerjack" to promote The Frightened City (plus a reprise of F.B.I.).
5. What was the first piece recorded on the rosewood-board Strat?
6. What was the last piece recorded on the rosewood-board Strats?
The answers to those two are bound up with the answer to number 4 (whatever it is).
7. What was the first piece recorded on the Burns?
Some folk say it was Geronimo, recorded December 1962 (that would have been a prototype). Others say Atlantis (same month).
8. What is the clicking sound in Jet Black? (The consensus seems to be Hank tapping on the pickup of his Antoria. Someone with an Antoria will please experiment.)
The consensus is correct. Hank is on record as explaining it.
9. Cliff used for a Chinese drum for Apache. What did it look like? Why is it so low-pitched? (Chinese drums are high pitched.)
See my answer in a nearby posting.
10. Was that a whistle in the Frightened City? Who made the sound?
Unlikely, I'd say. The explanation I was given when I bought the record in 1963 was that it was contact noise on the strings (the sort of thing that players use flatwounds to avoid).
11. Who played the bongos (or other percussion instruments) in Little B? The claves or other sticks? (I think Jim Nugent recalls that the rest of the Shads played Latin American percussion while Brian Bennett played the piece at a 1962 show in Newcastle that he attended.)
See my answer nearby.
12. Was the odd pickup pole[piece stagger on #34346 truly a discernible ingredient in That Sound?
I doubt it. Hank has always sounded pretty much the same with any stagger on pickups, and even with pickups with no stagger at all.