"I went to to the store for a newspaper" (there's no verb after for so the problem of the -ing suffix doesn't exist)

"I went to the store to buy a newspaper" to buy expresses the reason.

"I read newspapers for the news" What is the purpose of a newspaper? To read the news.

"I use newspapers for lining a birdcage" Q: What do I use a newspaper for? What is its function? A: To line the bottom of my birdcage.

"I went to to the store for a newspaper" (there's no verb after for so the problem of the -ing suffix doesn't exist)

"I went to the store to buy a newspaper" to buy expresses the reason.

"I read newspapers for the news" Why do I buy a newspaper? To read the news.

"I use newspapers for lining a birdcage" Q: What do I use a newspaper for? What is its function? A: To line the bottom of my birdcage.