1、overseas和for battle都作状语修饰谓语 were needed；
2、logging前面的in all不能单独看；它是一个介词短语in all logging,意即“在全部飞行记录中”；,in all logging over 60 million miles in all types of aircraft.意思是“（女飞行员）驾驶各型飞机的总飞行记录超过60000000英里”；
3、With fewer than 300 of the pilots still alive的意思是“有不到300名飞行员还在世”,fewer than用于限制、修饰300 of the pilots；你可以把 fewer than 300 of the pilots 看成是相当于“名词/代词”的一个词,就好理解了；
4、It was a secret trip the White House says was in the works for more than a month这句话,我们给它加上标点符号可能会更好理解
至于flying into Bagram Air Base for a little more than four hours to salute the men and women,这是两个非谓语动词短语——一个是现在分词短语flying into Bagram Air Base for a little more than four hours,修饰前面的：“It was a secret trip,”、"(it) was in the works for more than a month.",表示伴随状况,可能是记者在做报道上,一行人的飞机正在飞行途中；也可以这样理（因为前面有 the White House says,用的是现在时态 ）,所以flying into Bagram Air Base for a little more than four hours也可以理解成是（it will be） flying into Bagram Air Base for a little more than four hours这样一个完整句子的省略.it指“一行人乘坐的的飞机”；
to salute the men and women是动词不定式短语作状语,表示此行的目的.
希望我的解释能使你感到满意.