A gerund is the noun form of a verb. This allows you to treat an action as a thing. Yet, your examples have problems. Basically, adding "-ing" to a verb does not make it a gerund.
1) This is an incomplete sentence. Where is the verb?
2) This is an example of a present participle, not a gerund. Have closer look, and you'll see this is a condensed sentence.
3) This is simply wrong. You must use "to buy"
So here are the proper gerund forms of your examples:
1) Providing foods from different countries for us is the priority of ABC restaurant.
2) Hoping you will succeed in the exam motivates me to help you.
3) Buying this thing will please me.
Note all the examples use the -ing for to connect with the rest of the clause and stand as a thing in the sentence (yes, I've made all the gerunds into the subject).