The Commercial Press' 现代汉语词典 and many places on the internet say that 一下 and 一下子 are the same thing/meaning. But that's false, since the biggest use of 一下 is as a measure word and 一下子 is an adverb. Nevertheless they do seem to share some same uses/meanings.
I did some research on dictionaries and on the internet and this is what I've found:
一下子: at once; suddenly; immediately; momentarily; a little bit
一下: (measure word: quick actions, trying); one time; a moment; immediately (?); suddenly (?)
I'll give some sentences in which I think each meaning is very definite and the translations I've made (all examples are from the internet):
斧头一下子把木头劈开了。 The axe split the log AT ONCE. (in one blow).
他们把那只火鸡一下子吃光了。 They took that turkey and ate it all AT ONCE. (in one blow)
太阳不会一下子照及篱笆的两面的。 The sun can't shine on both sides of the fence AT ONCE. (at the same time)
她一下子就翻了一个跟头。 She SUDDENLY made a somersault. (unexpectedly)
我一下子想不起它叫什么。 I MOMENTARILY didn't remember his name. (for a short time)
我在人群中一下子把她认出来了。 I recognized her IMMEDIATELY in the crowd. (as I saw her)
他不是故意的，你就忍耐一下子吧。 He didn't do it on purpose, be A LITTLE BIT patient.
这一下子我可完了。 This TIME I'm screwed. (didn't find anywhere)
要是有可能，用左拳给他一下子。 If there's a possibility, use the left fist to give him A BLOW. (a punch, didn't find anywhere)
现在我们把条件降低一下。 Now we take the conditions and lower A BIT.
我们停下来让马休息一下。 Let's stop and let the horse TAKE A rest.
他的脑袋撞得这一下很重。 His head bumped so serious this TIME.
如果我们能借用这个一下陛下。 If we can borrow this for A MOMENT, Your Magesty.
她听了扑哧一下笑了起来。 She heard a "splash" and SUDDENLY/IMMEDIATELY started to laugh. (which one? unexpectedly or right away?)
那鱼一下子咬住了鱼饵。 That fish bit the bait AT ONCE/SUDDENLY/IMMEDIATELY.
1 - Are the translations for 一下子 (time, blow/punch) right?
2 - Are there any more uses for 一下子 that I missed?
3 - Are there any wrong translations for 一下子 or 一下?
4 - What's the best translation for 一下 on the last example?
5 - What 一下子 examples could I use 一下 as a substitute?
6 - How would I solve the "bonus" translation?