The use of wa after nakushite indicates that the sentence is a conditional sentence. This means that when a sentence has a conditional meaning, wa is mandatory as shown in 山川教授... = perhaps nakushite is "without" = perhaps nakushitewa is "if it weren't for" - slightly stronger
私の許可なくして is perfectly valid in terms of written Japanese. In terms of spoken Japanese, perhaps 私の許可を受けないで would be normal, even though 私の許可なしに would also be acceptable.
みきさん、 Below is some explanation from my 先生： 「私の許可なくして＿＿＿」の文は日本語ではあまり使いませんから、 正しい文は「私の許可なしで＿＿＿」でしょう。 or 「私の許可なしに、______」
ex #5309 would mean that if it wasn't of my permission, you shouldn't have done such a thing. I am actually doubting the usage of なくして in that example, not the Japanese. Instead I suggest: 私の許可なくして、君はそんなことを達成するのが無理だ。 If it wasn't of my permission, it is impossible you could achieve such a thing. After なくして、 it is natural to be followed by something that one can't achieve or something that is impossible to happen without it. In #5309, it is 'shouldn't' instead of 'can't/impossible' Here are some examples from a book: 1.彼は、人工呼吸器の助けなくしては一日も生きられない体だ。 2.日々の努力なくして、成功はあり得ない。 3.それは、涙なくしては見られない感動的な映画だった。 ４．皆さんの応援なくして、優勝はなっかったと確信致します。 ５．苦しい練習なくして、試合に勝つ事はできあないのだ。 ６．悲しみなくして、人生の喜ぶを知ることはできない。 ７．如何にお金があろうと、育てる者の愛情なくして、子供は幸せになることができない。 There is a difference between 'If it wasn't for...' with 'without'.
As for #5309, I don't think the Japanese is wrong even if it sounds strange when translated. Please advise a proper translation for it.
I agree that it should be translated as " If it wasn't for...". By the way, in ex #5309 the usage of なくして seems strange. Because it sounds strange when translated, [ If it wasn't my permission, you should not have done such a thing]. 〜なくして（は）．．．ない where the usage usually comes with "If it wasn't ..., then..(followed usually by sentence stating : can't / doesn't have the potential or possibility to do or have something)" It should have been : 「私の許可がなかったら、君はそんなことをすべきではなかったのだが．．．」 If there wasn't any of my permission, you shouldn't have done such a thing... Or 「私の許可ではなかったら、君はそんなことをすべきではなかったのだが．．．」 If it wasn't of my permission, you shouldn't have done such a thing. Or another example for なくして, 皆さんの協力なくして、そんなに辛い事を解決できないのだ。
I think a good translation for this is "If it wasn't for_,".