(2) There is one major difference between the situations of Arthur and Obama: Arthur was beyond doubt a citizen of the United States at the time he assumed the presidency; in contrast, we still don't know if Obama is a citizen or even if he ever has been a citizen. (I admit that this has little bearing on Arthur's constitutional eligibility or lack thereof).
(3) As far as American troops saluting the British flag is concerned, my guess would be that, given the close proximity and working relationships between Americans and British in WWII and the stationing of many American troops in Great Britain, it possibly did occur during that time frame, but probably not on orders of the president.
From what I've read and heard, standing at attention before a foreign country's colors almost certainly was common then (and still is). However, to actually salute the foreign colors, less certain. IIAC, saluting the "colors" is a sign of not only respect, but shows allegiance.
Of possible interest...
"§ 9. Conduct during hoisting, lowering or passing of flag
During the ceremony of hoisting or lowering the flag or when the flag is passing in a parade or in review, those present in uniform should render the military salute. Members of the Armed Forces and veterans who are present but not in uniform may render the military salute. All other persons present should face the flag and stand at attention with their right hand over the heart, or if applicable, remove their headdress with their right hand and hold it at the left shoulder, the hand being over the heart. Citizens of other countries should stand at attention. All such conduct toward the flag in a moving column should be rendered at the moment the flag passes."
Patrilineage. The citizenship of the husband determined the citizenship of the wife. The citizenship of the father determined the citizenship of the child. Chester A. Arthur was not even a citizen at birth, regardless of where he was born. Regarding a wife's citizenship as an entity separate from her husband, only came into being in the 1920's.