Keep giving her the foods you eat. She will develop a taste for them over time.
Feed her things like strawberries, and other fruits, cut up of course. My little ones always liked canned pears, and they are nice and soft, too. Vegetables like sweet potato, mashed potatoes, finely shredded chicken breast, cooked carrots, vanilla yogurt, scrambled eggs, etc.
If you eat foods that are really rich in flavor or spice, she may not be ready for those yet. Make her foods with little or no salt, pepper, etc. Let her learn to appreciate simple foods.
Tomatoes were also always a big hit with my kids.
As for the cup issue, sounds like she is skipping a step. Have you tried the sippy cups with the soft flexible lids? They feel less foreign sometimes. Some kids will drink out of the spill proof, while others won't, perhaps because they are harder to suck the drink out of - they have to suck harder.
Try a variety of differnt kinds of cups and see if that helps, but other than the inconvenience of eagle-eye supervision and help with a regular cup, there certainly is nothing wrong with her using a regular cup instead, if you have that kind of time...
Good luck.
It sounds like she is really developing an independent personlity.
Congratulations. Perhaps she will be president - and show the world her way of doing things is better.