しょうがない on its own is a sort of idiom meaning "Can't be helped," but because this is a grammar entry, it should be emphasized that しょうがない is correctly used in a sentence when it follows て or で. So, てしょうがない is a proper grammar entry title. In that case, example #939 is simply a conversational use of the idiom and doesn't belong in this entry.
shouldn't it be "shiyo ga nai", not shou ga nai,
Author: your name
According to Daijirin, both ways are ok.
The title should be "shiyou ga nai" and て should be deleted. I have added an example. NOte that しようがない is commonly pronounced as "shoo ga nai."
#5458 in Japanese is weird. Also, in #276, "tired" should be "sleepy."