Looks like a nice cage. The front shot of his beak is normal. Is like to see another side shot that shows the bottom and sides of beak. Bet yes the peeling of the top beak is complete normal.
A plate wonāt feel safe to most birds for taking a bath. What works for me is a glass square 9x9 casserole dish ( brownie making dish

. ) I also have a shallow serving dish that is very sturdy that I use for the baths to. Only full about an inch of water then splash splash with your hand he should be interested leave him to explore it.
Veggies keep offering! Expand what he will take Denver they may reject it the first five times you offer it then one day itās their favorite veggies ever! South America parrot species like Quakerās eat peppers in the wild, do they usually take to peppers very easy. They canāt taste the āheat ā of peppers and so even seem to like hot peppers best. Mine like the red chili peppers best of all but eat bell and other peppers to. The red chili peppers are close to the wild type peppers so that might be why they are liked best. Mine like romaine lettuce, zucchini, broccoli, Brussels sprouts, fresh green beans , peas frozen thawed or fresh never use canned anything, sweet potatoes they only like cooked,
For your toys what you have is fine you just need to add stuff that is easy to tear and chew up, they sell this woven grass like stuff called shredders they love to rip that up, Aldo that stuff at the bird stores that looked like rolls of cardboard tape they like to destroy those, and those soft yucca chips ( I think that what called) they love those to. The birds like to effect change on the environment and they like to take stuff apart and destroy it so some of their toys need to be that type . You can DIY stuff to

old paperback books give them to the parrot to destroy it weave it in the cage, cut up cardboard into fun shapes pinch a hole through it and use a zip tie to dangle it near a perch .
For perches I like to give ten more of a jungle maze, and use some shirt wide perches placed around with yummies and toys by them to encourage use of every square inch of the cage. Observe your parrot and how uses the cage and adjust and move stuff according.
You could hang we
One of those big swishy bungee ropes from the top center of cage so it hangs all the way to the bottom of the cage. One of my Quakerās like ladders ( tho I have one that doesnāt)
Also you can attach perches on the outside sides and top of the cage! Itās great turns it into a jungle gym! Things donāt have to be straight across birds live climbing , live hanging off stuff, love to see if they can jump from one perch to the next without flying they are winged monkeys !

I also use the vertical space above the cage by putting in ceiling hooks and using fishing line to hang this big rope hoops down just a few inches above the cage top and to Han a couple of swings
My safe bird store has lots of great stuff! Check them out