To sleep is usually understood to mean to be in the state of being asleep, not to fall asleep. Because sleeping is usually a long event (eight hours is commonly quoted), stating that you were asleep at a particular hour or a specific time makes little sense, unless you also state that you only slept for a short amount of time.

to sleep: to be in the state of being asleep

to fall asleep, to go to sleep: to change state from being awake to being asleep

to wake, to awaken: to change state from being asleep to being awake

to be awake: to be in the state of being awake

                awake             asleep
be in state     to be awake       to sleep
change state    to (a)wake(n)     to fall asleep

So sleep is not the opposite of wake. Asleep can be either an adjective or an adverb, sleep can be either a noun or a verb. I'm not aware of any single word verb for to fall asleep e.g: to go to sleep, to pass out, to lose consciousness, to become unconscious, etc. Here sleep and wake are used as verbs, while asleep and awake are used as adverbs with another verb to make it grammatically correct. Note the lack of symmetry!

When did you fall asleep? I fell asleep at 10 pm.

How long did you sleep (for)? I slept for eight hours.

When did you sleep? I slept from 10 pm till 6 am.

When did you sleep? I slept last night. (A long enough period of time to complete the process)

When did you sleep? I slept at 10, 11, 1 & 4 but I was awake the rest of the time.

In the sentence I ate at 10, eating usually takes less than an hour, so it is not saying that you were eating for exactly one minute at 10 o'clock but that you started eating at some time close to 10 and continued until you were finished a short time later, whether that took you five minutes or half an hour. Both I ate at 10 and I slept at 10 imply that you weren't doing the same activity at 9 o'clock or 11 o'clock.

Please note that Indian English uses words and meanings considered non-standard in other varieties of English such as prepone as an extension of postpone. It seems that these words and meanings crop up because Indians assume a symmetry in English that does not usually exist. They can be usually be understood without further explanation but cause initial surprise when encountered. Since you said that I slept at 10 is common in India, keep using it when speaking to Indians. Just be aware that not everyone outside of India will understand that you mean I fell asleep at 10. So most people will understand you even if they would not say it that way themselves.

It is to do with the amount of time it is expected for the activity to finish.

I drove to work at 8. I arrived at 8:30. I worked at 9. I slept at 10 but luckily I woke at 10:30 before the boss saw me at 11. I ate at 12.

drove - continuous, less than an hour is common

arrived - a single moment

worked - continuous, less than an hour is uncommon, up to eight hours is common, sometimes (a lot) more

slept - continuous, less than an hour is uncommon, eight hours is considered normal

woke - a single moment

saw - may be either a single moment or continuous depending on context

ate - continuous, less than an hour is common

Note that my story is very fragmented, and not just because it is using very short, simple sentences. It's not clear what I was doing between 8:30 and 10 except at 9 or between 10:30 and 12 except at 11 because I only said what I was doing at specific times. It is assumed that I was driving for half an hour and that I slept for half an hour because they are both continuous activities and the next event in the story concludes each of them.