Why don't you like drinking beer?
Is it correct to use ING in the question above?

I know that we can say "do you like drinking?" and "don't you like drinking?". But can we use ING when question comes with a question word?

Examples:

What don't you like eating? I don't like eating chocolate. (I do)
Who wouldn't you like meeting? I woundn't like meeting Hitler.
Where didn't you like going? I didn't like going to camp.

See? When we are asking a "negative question" using question words and verbs like "like", "love", "hate", "start", "quit", "finish", "don't like" and other verbs, something sounds wrong.

By the way, when can we use ING after verbs?

Question

Why don't you like drinking beer?
Is it correct to use ING in the question above?

I know that we can say "do you like drinking?" and "don't you like drinking?". But can we use ING when question comes with a question word?

Examples:

What don't you like eating? I don't like eating chocolate. (I do)
Who wouldn't you like meeting? I woundn't like meeting Hitler.
Where didn't you like going? I didn't like going to camp.

See? When we are asking a "negative question" using question words and verbs like "like", "love", "hate", "start", "quit", "finish", "don't like" and other verbs, something sounds wrong.

By the way, when can we use ING after verbs?

Answer

In my opinion, all your examples sound fine.  I think the reason they sound a little bit awkward is because we wouldn't necessarily use the contraction.

"What do you not like eating?"
"Who would you not like meeting?"
"Where did you not like going?"

These all sound more natural to me.

You could also use the infinitive form instead of the -ing form.

Answer

For the second question, a simple online search (verbs followed by ing) will give you plenty of examples. Here's one:

http://www.englishpage.com/gerunds/gerund_list.htm

You just have to learn them.