Rice has high levels of sugar so I would definitly avoid it ---especially if he is taking it over other foods. Nutritional problems can cause behaviors and if rice is filling him up, he may not be hungry and he may be missing vital amino acids etc.
If you must give him rice, I would use it as a reward and give like 2 grains at a time (cooked). That could be a good cage bribe.
Okay, so now we know why he hates his cage (he was locked in it all of the time).
He is anxious about it because he had bad experiences there and doesn't want that to repeat.
When I got my cockatoo, she eventually took food from my hand, but it took a month or 2 after that for her to step up willingly, so I think that can be normal with trauma. Yours may move very slowly because of past incidents.
It is a good sign that she likes you enough to take food---
she is probably scared of hands because A- They hurt her at some point, or B- she was locked in her cage and didn't get exposed to them much.
You could try taking the food out at night and just see if that helps. If he isn't eating during the day after not eating that night, then you may want to put it back in, but he won't starve to death in 24 hours. If motivation is increased by hunger, then no more night eating might strengthen his resolve to get food. Talk to your vet about how long you should wait before caving in.
When you let him out of his cage, it sounds like he is exiting on his own. AFTER you solidify his trust (via time+positive associations)you might consider opening the front door and setting a hand near it. If you can get him to come near your hand, then that is progress. I am not saying to force him to step up, but you might see if he will even exit with your hand nearby. Keep your hand there and see if he will walk past- do not move your hand. Eventually, he has to learn that hands are not a threat, but this will take a long time. Gradually, move your hand closer to the door and closer to him- before opening the door (and only after solid progress is shown). You could work towards having him step up before he gets out (stepping up= what you want, getting out= reward). Always pair the reward (letting out) with consistent verbal praise. This will come in handy in the distant future.
You will need to break down each step in the process via a "task analysis"----reward and shape baby steps and then stack them together over time. During teaching, you reward any approximation toward your goal (if he approaches your hand, let him out. When this is consistent, add that she has to touch your hand, then let him out. When this is consistent, add that he has to approach, touch and put on foot on your hand, then let him out. Getting let out is the reward, and stepping up has multiple "steps" lol so break them down and then teach them in sequence.
Before you do any of this though, I would give him time and I would close off his play area somewhat (smaller room, tension rod+sheet, door ...something). Think of a space the size of an office maybe. Make sure windows are covered or have decals on them so that he can't slam into them if frightened-- look for hiding places and if you can remove them without terrifying him, do so. Initially, I would just sit next to the closed cage and do quiet activities (laptop, reading etc). When you let him out, just sit in there with him- and ignore him. Keep doing what you are doing and just stay calm while he feels out the situation. Do this for multiple days and aside from occasional quiet talking, let him initiate any interaction. With limited stimulation, he may get bored or hungry and decide to go back in to get food etc. When he goes back in on his own, give her a food reward in a non-scary way, but don't close him in. I would wait for him to get out and then let him see you put a favorite treat in his cage bowl. That way, if he does decide to go back in, you don't have to scare him by walking over (OH NO, SHE'S LOCKING ME UP---PANIC PANIC!). Let him get used to coming and going with you in the room and show him that you won't always lock him up for going in his cage. If he likes toys, you could also try putting them in there, but often birds are scared of new toys.
Those are my thoughts initially....Make sure all people who interact with him follow the same protocol/plan with regard to their behavior and actions as well. One person could de-rail progress by scaring him or rewarding bad behavior inadvertently. You need to prove to him that he will not be locked up every time he enters his cage and you need to teach him that it isn't a punishment to go inside of his house. Once he is able to step-up etc, you can gradually thin out rewards etc and expand his play area, but right now, it is too much freedom and chaos (which is feeding his fear via nightly cat-and-mouse routine).