We do have another cat who is older and a bit of a bully. They have a kind of live and let live relationship but if the little one gets a bit too frisky the big one puts her in her place.
I'm thinking it's just general kitten naughtyness at the minute, jumping on counters and stuff like that. We can handle pushing her away when we're eating but the early morning faffing and scratching the doors is annoying. It would make sense if they were closed and she wanted to come in but more often than not, they're open. Something to deter the door scratching would be ideal.
She's also being a right bugger about eating the bigger cat's food, which isn't ideal the big cat is on a strict diet with special food from the vet - not cheap! We've tried moving it, putting it on shelves, behind stuff etc yet you'll always find her munching away. It's kinda cute, but annoying!
I'd be pretty certain that what you have here is a territorial battle. By and large, cats hate other cats. Your kitten has grown up enough to begin the process of wanting to usurp the older cat and the older cat knows this - hence trying to bully the little one while she still can.
The scratching is territory marking, and because it gets your attention, has the benefit of making her the 'special one'. The older cat, by default, is therefore marginalised. This is likely to be the case with the food stealing too. Aside from the fact many 'special diet' foods are tastier than normal, the kitten raiding the older cat's food sends the message: 'Not long now, I'm in control oldster'.
You note that you find this 'cute' - which it is, but just as with naughty children, the kitten is getting all the attention for behaviour you actually want her to stop. Similar with 'pushing her away' when she tries to eat your food - not harsh enough to send the right message. I'll bet this is done with a smile and a giggle until she finally winds you up enough to say no. This sends all the wrong messages, but we're all guilty of it with a cute kitten.
You need to give the cats separate territories as much as possible. Set aside a room if you can for the older cat to have its bedding and or food. She should be fed behind a closed door, and first, so she can finish her special food. She goes first because she is the territory holder and should be treated as the dominant cat (until they work out the hierarchy themselves, once the kitten is grown). The kitten should not be given special privileges, and should also be fed in her own territorial space, different from the first. They should have separate litter trays - and if they go out, you should let them out at different times - again, with the older cat going first.
The kitten needs to be trained to respect the rules of the house and not indulged. The older cat needs to be petted and fussed as much as the kitten. The scratching should be punished with a scolding and if necessary, with a short spray of water. Good behaviour (i.e. scratching the appropriate post) should be rewarded with petting, and sometimes with a treat.
Don't expect your cats to get on. Even siblings tend to dislike each other most of the time. What you see as a 'live and let live' relationship is actually cold, bitter warfare. A newly introduced kitten with an established cat is the worst of all worlds. Just think how irritated you would be if someone set up home in your house with nary a 'by your leave' and promptly started eating your food and getting hugs off the spouse in preference to you.
I would also invest in the Feliway noted previously. This spreads a cat pheromone that communicates to all felines that the world is mellow. Along with the above, it will help smooth frayed nerves.
Best of luck!