sleep和sleeping在这4个句子中都对吗?

从语法来看这些句子都对,
但第一句这样说比较地道：
Are you recently unable to sleep?Or do you have sleeping problems?（省略 some)
或
Are you recently unable to sleep?Or do you have trouble sleeping?
Do you have sleeping problems?和 Do you have any sleep problems?意思一样
第四句 keep 不是最适当的动词,改为 maintain 比较好：
So how can we maintain good sleep in the fast-paced and stressful environment?（不加冠词 a ）
而这样说更地道：
So how can we get a good night's sleep in the fast-paced and stressful environment?
我是加拿大人,前英语老师.

正确。

Are you recently unable to sleep? Or do you have some sleeping problems?
better: Are you recently unable to sleep? Or do you have a sleeping problem?
Do you have any sleep problems recently?

sleeping的意思是用来描述和住宿有关的、或者睡觉的地方和睡觉的用具的，如sleeping tent, sleeping bag, sleeping arrangement
睡眠问题是不应该用sleeping的，应该是sleep problems.

所以第一句的后半句不是很准确，其它的都是对的。当然，因为用sleeping problems也不会引起歧义，所以这么用...