When you describe the action, which ones are used, and what does the other pair's usage and meaning (a + verb), if they are unfit?
- This is a walk.
- This is walking.
- This is a buy.
- This is buying.
- This is a sleep.
- This is sleeping.
This is walking.
This is buying.
This is sleeping.
There are few contexts I can think of where using these sentences would be appropriate. The two that I can think of are:
Something said as part of a demonstration or example. Perhaps you are explaining a word to a non-English speaker. You walk, and tell them that "this is (called) walking."
Something said to emphasize that "this" is a particularly good example of "-ing." A common example of the phrase being, "(now) this is living," which means that the speaker considers their current lifestyle to be a good one, and probably one they wouldn't change.
This is a walk.
This is a buy.
This is a sleep.
As @P.E.Dant said, these are all nouns (not verbs), and #3 is incorrect. Generally, sleep is an uncountable noun, although it can be countable in certain contexts. For example:
There are two sleeps until Christmas.
Meaning: There are two nights left until Christmas day.
I can only imagine this is a walk being said in the context of a sponsored walk, when somebody asks what you are doing: "This is a walk (to raise money for charity)."
This is a buy is slightly more applicable. It's quite common to describe a low price for a product as a good buy. The term is also used for drug deals, as in "it's a (drug) buy".
I'm not certain that I've completely understood your question, but I will try to answer it.
If you are trying to describe what you are currently doing, you would say:
I am walking.
I am sleeping.
This can also be used to describe future actions:
I am buying a house (next Tuesday).
Be careful when using buying, as in certain contexts you may want to use shopping instead. Compare their definitions (from Oxford Dictionaries):
Obtain in exchange for payment:
"she bought six first-class stamps"
"he had been able to buy up hundreds of acres"
[WITH TWO OBJECTS]: "he bought me a new frock"
[NO OBJECT]: "homeowners who buy into housing developments"
Visit one or more shops or websites to buy goods:
"she shopped for groceries twice a week"
"sometimes it's more convenient to shop online"
[WITH OBJECT]: "take a trip to downtown San Diego to shop the upscale stores of Horton Plaza"