I am tempted to put large empty box from ALDI to cover the top of refrigerator. Does the box need to be really clean, or if it's appear clean would be enough? Could it trigger nesting behavior?

If it looks reasonably clean that's all you need. An empty box might trigger nesting behavior so I'd suggest filling it up with something inconvenient. Wadded-up newspaper might work if the surface is so unstable that the birds can't walk around on it.

How does grit work? I understand that it helps with digestion but why should it be spread on the cage floor? not very sanitary I think. I got a box of Hartz grit at walmart.

I meant boxes with open front. it's Ok if birds get in and chew if they want. i just need something on top of the fridge to cover areas where they can fall into gaps between fridge and walls

Grit doesn't have to be spread on the floor, you can put a small amount in a dish if you want to.

You'll have to watch and see how the birds treat the box. If they act like they think it's a nesting area, you'll have to do something to convince them that it isn't.

Same baby again. Today I noticed she has red in her beak (when I put my finger close to her beak and she opened her beak, it looked like she has a little bit of blood there. not sure what it's called... analog of a tongue). Probably got hurt on something sharp in the outdoor area. I will need to check if there are any sharp wires. She was acting all right but later I noticed that her nostril dots looked kind of reddish... She is out of the cage, I tried to catch her to examine but she escapes. I will go get some cornstarch... I am really worried.
I Looked closer. Only one of her nostrils is red. But what is interesting - her cinnamon twin sister also has one red nostril. They are both alert and flying all right.

they also play with the toy that has dyed sticks and wicker balls

They play with dyed toy... Maybe its coloring from the toy?

I do hope it's the dye... Let us know !!

Staining from toys is very common. The dyes are nontoxic food coloring and they're not very colorfast.

The cere can also look red if the bird has just been flying a lot or exercising in some other way, just like the way our faces get red from exertion.

You want to see my birds after eating beetroot!