"I am withdrawing myself from the situation" - вынимать is not the best word here. Вытаскивать is better in this particular example. Also, neither has have a reflexive form (well, you can make it, but it would sound weird or/and funny 8) ). "Я вытаскиваю себя из этой ситуации" sounds more Russian. We are kinda dragging or pulling ourselves from tricky situations.
Вытаскивать
"Я вытягиваю себя из этой ситуации" - also possible, more slang.
"Я выбираюсь из этой ситуации" - an example with reflexive verb.
"Я выхожу из этой ситуации" - an example without reflexive words or endings
"Я вылезаю из этой ситуации" - more colloquial/slang
"The government is 'withdrawing/pulling out' of the war" - in this example, "выйти из войны" could work, especially if you mean a country. http://wordweb.ru/history_ww2/50_02.htm the title of this article is "An attempt for Finland to pull out of the war". Your example would be something like: "Правительство вышло из войны". Though, it sounds a bit strange to me, because we usually mean the country pulling out of the war, not specifically it's government. So, with a loose translation, I would say "Страна вышла из войны".
"I cannot express myself correctly in Russian" - both are possible, though it is better to use a reflexive verb here, "выражать себя" - is a bit lyrical. "Я не могу правильно выражаться по-русски/на русском".
A little remark on "can" here. You may say "Я не умею правильно выражаться по-русски/на русском" and it would mean that you cannot express yourself correctly because you haven't learned how to do that yet. This would sound a bit better because that way you suggest that you are learning Russian and trying to express yourself correctly, and that gives you more respect. You can add "ещё/пока ещё" (still/for now) to make it even more naturally sounding. "Я пока ещё не умею правильно выражаться по-русски" - sounds pretty natural.