No way! No offense there buddy, but that just isn't right.
One of the reasons she isn't getting as much screen time is because Spike at Your Service was meant to be her episode, but they thought she was acting mean. This means two things,
1. The writers wanted Rarity to get an episode, but had to cancel it
2. They cancelled it because they thought she was being too mean.
They wouldn't have re-written basically a whole episode because they didn't think Rarity was in character if they hated Rarity.
Also, Applejack isn't stealing any spotlight. She is constantly referred to as a background pony due to her low screen time. If you mean on the good sister front, well of course she's a better sister. Rarity isn't good with kids; she still has a lot to learn, even after Sisterhooves Social. Applejack, however, knows a lot about being a good sibling, having lived with family her whole life and being used to that kind of thing. Naturally, she's the better sister.
Now, you're bringing Twilicorn into this. Of course.
Look, let's just say Pinkie Pie was a unicorn, and Rarity was an earth pony. Just hypothetically.
They wouldn't stop Twilight's transformation for Pinkie's sake, and that means the same for Rarity. Twilight was gonna become an alicorn regardless. And I'm kinda sick of people saying it's throwing off the balance of the group. The balance is fine. Does there really need to be 2 unicorns, 2 earth ponies, and 2 pegasi? Is it really that important?
Also, she is often seen as greedy and a bad sister for a but of comic relief. Also, her being greedy in Secret of my Excess was kinda necessary for the storyline. I guess Spike could have given her the ruby by himself, but it doesn't really matter that much.
Besides, I'm pretty sure Rarity would be a pretty hard character to write for.
So, now you're bringing in a fan fic, to explain why the writers hate Rarity. Yeah. That makes sense.
Look, I haven't read the comics, so I can't speak for that, but I've seen the show, and trust me, she's not hated. She's going to get at least one episode in Season 4, it's been confirmed.