Curate, connect, and discover
Rouge is absolutely, out of all the Sonic cast, the one who collects memes/videos/posts that she finds funny or that remind her of her friends.
Rouge shows Shadow mostly self-help posts and bits of poetry. She also knows that he likes those cheesy "tough guy" memes, especially when it involves guns or motorcycles. He likes to print out any posts that resonate and hang them up in his room.
Rouge shows Omega any and all posts or memes about guns and explosions, of course. But she also finds and shows him the best of those "epic fail" videos- seeing meatbags doing stupid shit and falling on their faces for it gets him every time. She's sworn to never tell.
Rouge sends Sonic memes people make about himself, mostly. But occasionally also a meme so damn abstract it transcends the concepts of space and time.
Rouge and Tails scour the internet for the worst knockoff Sonic merch they can find. It's a competition between them, honestly. Rouge is currently winning after she found the world's worst AI generated mug, where it looks like the intended idea was for the user to be drinking their beverage. . . out of one of Sonic's shoes. (She keeps threatening to buy it and send it to his workshop whenever he annoys her.)
Rouge and Amy don't send each other anything, but they do share a few pinterest boards and collaborate together on playlists. The one exception to this was a meme Rouge found and just had to show- someone had managed to snap a photo of a Badnik in a convenience store and captioned it with "robit in da grocery store, what he gonna buy". Amy laughed so hard she cried.
Rouge almost exclusively torments Knuckles with the most cursed images known the man. It's similar to that old bit from Peanuts- she's like Lucy pulling the football away from Charlie Brown at the last second. This time, she promises, she is showing a funny joke. She promises! (it's all in good fun though. Little does she know that Knuckles collects the cursed images to send to Sonic whenever he gets annoying.)
Rouge's peak achievement was getting the whole gang together for a dramatic reading of a terribly-written real-person fanfic about Eggman. It was horrendous. Sonic just about stopped breathing from laughing so hard. Omega threatened to burn the apartment down seven times throughout. And Tails? Tails made it into a copypasta to spam in Eggman's inbox.