These weird wording have been causing headaches for a year now, with not every card being clearly explained. When doing my research for my FAQ I found that the wording for Barkman had been verified by HQ members.
I will try to re-condense all cards (OFFICIAL) status:
Global:
Sharpshooter gunner: (The rule you described). Unclear (at least I didn't find an HQ statement on it). However our group is using the ruling for Ernst Barkmann here: if not explicitly stated "first OF THE GAME", the intent is EVERY TURN. This is of course our house ruling for lack of better evidence, but it works well, and makes such cards more interesting. This will get sorted out with an official FAQ, or at least ANY responses from HQ which must've gone silent like 3 months ago and were never heard of again....
German:
Ernst Barkman: If he scores no uncanceled hits, he gets to make another (new) attack. Every turn that is, not just the first.
And yes, re-rolls do apply to both attacks (if applicable at all), for they are independent attacks, not just re-rolls of re-rolls. (this is also true for extra attacks by Bobby Woll / NOT Audie Murphy though, for he has to discard all other crew+upgrades prior to shooting, thus loosing any and all eventual re-rolls for his attack)
Dr. Fr. Ba.: makes a SINGLE arrow move at the beginning of the game, not full movement of the tank he is in.
Otto Carius: His -1 on targets armor and +2 initiative applies even when moving, as in the sentence with being stationary it is clearly stated he gains an extra attack from not moving. Rules stated go sentence by sentence as many other cards show.
USA:
Wardaddy Pool: Unclear how applied. heard many play it as "first of the game", whereas our group rules it, again, like Barkman: unless stated otherwise rules are applied every turn.
Ground Hog Oller: Unclear how applied. Our group rules it, again, like Barkman: unless stated otherwise rules are applied every turn.
England:
Wilfred Harris: Unclear how applied. Our group rules it, again, like Barkman: unless stated otherwise rules are applied every turn.
Russia:
Maria Oktyr.: Changes the (moved) tanks base-armor to two (2), DOES NOT ADD two (2) armour to the tank the is equipped on. Thus an IS-2 will, even when moved, remain at armour 2, whereas an T34 (the tank she drove in WW2) would now be driving around with armour 2.
Vasiliy Kryshov: Gets one (1) defense-die-re-roll per move made in that respective turn. This means for the whole phase, not every instance of being shot at. Example: He is driving a T34 (fast) and moved 3 times. He gets shot at and hit 4 times, cancels 1, now he has 3 re-rolls he can use on this shot (if desired). If he uses all three and gets shot at again (eg 2 uncanceled hits this time), he now has no re-rolls available for his defense-rolls. He moved three times and gets three re-rolls, to be distributed as desired. but if shot at 15 times he cant re-roll 15 times.
Kommissar: Unclear how applied. Our group rules it literally: He himself counts as a Crew-Upgrade, so does AMMO etc., therefor he can shoot himself or even the ammo on board to re-instill morale. Sounds dumb, but that's how the card reads.
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PS: Generally my group comes from a rather competitive background in other games, therefor we interpret cards not logically, but literally. If it says A it means A, no matter what intent B is. Cards are in the game to reflect an intended rule, not fluff-text. A blatant example is the card Bailed Out, as of right now Bailed Out tanks can still hold objectives etc, not very logicall, but the trick is to read the card as its rules (cant move, cant shoot), not as its name "bailed out". This is not to dictate how you should rule, but to explain why/how we derive our rule-interpretation.
This, as any other ruling, needs an official statement though. I marked unclear rulings in bold, however you want to interpret them is up to your group until official clarification arrives. Further I want to add, that our house rules are not water-tight, as EG "best job I ever had" clearly states "every/each turn" (rather than an exempt-line like "of the game"), therefor these cards can be interpreted however. We stuck with the general Barkmann ruling, because that is the once instance HQ actually responded and clarified that it applies every turn unless stated otherwise.