Yes, they are both correct. In "without sleeping," "sleeping" is a verb form and refers to the activity. "He is sleeping. He is lying in his bed, and he is snoring." In "without sleep," "sleep" is a noun. It is as if "sleep" were a kind of stuff, and substance, which we can get or have or catch. "I got seven hours of sleep last night." "I wasn't able to catch any sleep." The only way "to get sleep" is "by sleeping." Sleeping is the action, from which you "get" the noun. For this reason, "she went without sleep for twenty hours" logically means the same thing "she went without sleeping for twenty hours." Similarly, "he went without food for a week" logically means the same thing as "he went without eating for a week." Different words, different grammar, but the same meaning because of the real-world knowledge of "food" and "eating."

They're both correct yet the construction of your sentence would depend of which one to use. "Without sleep" is using sleep as a noun. "Without sleeping" is using sleeping as a verb (present participle).

Yes, they're both correct.

Good question. It all depends on context. You can say "I can go without sleep for days" or "I can go without sleeping for days" and it would be correct. However, you could say "Without sleep, a person cannot function well" but it would be awkward to say "Without sleeping, a person cannot function well". Hope that helps.