(a) He is able to name things in the home.

(b) It's healthier to eat fruit.

(c) We have to eat healthier fruit.

(d) The king told the people to cut the ropes of bridge. It fell down.

(e) Get a glass of jam jar, a cover, a piece of cutter and a paintbrush.

good

b)

Good, but usually "healthier" (a comparative adjective) suggest you need to compare "fruit" with something else. So you would write something like "It's healthier to eat fruit than candy." One alternative is to use the positive form of the adjective (healthy) and say, "It's healthy to eat fruit".

c) 

Good, but most people consider all fruit to be healthy, so it would be hard to say one fruit is "healthier". However, if you meant "We have to eat healthier" and wanted to suggest fruit was an example of eating healthy, then the sentence would need to be changed. Example (you can use "healthy" instead of "healthier"):

We have to eat healthier food like fruit. 

d)

Good, but try to avoid pronouns like "it" as the subject of a sentence if they could confuse the reader. In your case, "it" cannot refer to "the king" or "the people" because "it" is used mainly for objects/animals. Also "it" cannot refer to "the ropes" because it is a singular word.

e) 

Good, but remove "glass of" since you already referred to the object as "jam jar."