Hi,
I'm not very experienced with IRN's, we have only had our mature boy for a few months now. But I can maybe answer a few of your queries, and hopefully someone else will chip in with some info for you also.
1) I've heard that can you mix in the pellets with the seed, start off with a small amount then gradually increase the ratio. You might also want to consider taking out the sunflower seeds from the mix, and only giving as a treat when rewarding or training your bird. Sunflower seeds are high in fat, apparently, so you won't want your bird just eating them alone
2) Training/Talking - I've heard stories of IRNs as young as 3 months trying to form words etc, and others say 7-8 months is when they start getting vocal, but it all depends on the time and effort you put in, I guess. It certainly wouldn't hurt for you to start trying now. IRNs are very good mimickers, and will usually start mumbling to themselves when they think you've left the room/house etc as a practice run. Just talk lots to him/her and maybe start of with short phrases like "Hello (name)", "Pretty boy/girl/bird", "Good boy/girl/bird" etc. Also, youtube is a great place to find vid's for training parrots .
3) Closing one eye - Is it possible he is having a nap, but keeping one eye open for predators? Birds (and certain other animals) actually do this - there is a name for it but it escapes me at the moment - they sleep with only their brain or something, one half shuts down but the other half is still awake, on patrol for potential predators, if you will. I don't think I'd be worried unless there were other symptoms present that might indicate he was sick.
4)Bathing - We use a squirty (spray) bottle to give Skittles his bath. We usually have him outside on the back patio for awhile first so he can warm up, then spray him through the bars of his cage. Before long he is usually hanging off the side of the cage, spreading his wings out and back, so we can squirt him all over......and then if we stop, he usually goes straight for his water bowl and tries to get his whole body into it! Hilarious to watch
You can also buy large stainless steel bowls that attach to the side off the cage for them to bath in, but I'd try the squirty bottle first, and see how that goes.
5)Getting IRN back into cage - When inside, we leave the door to Skittles cage open so he can come and go as he pleases. He has a 'playgym' made from a tree branch stuck in a bin on wheels covered in climbing ropes and toys that he spends most of the day on, only going back into his cage for food and water, and to sleep at night. Our bird isn't hand tame AT ALL, so if we want him to go back in before he's ready (if we need to leave the house for work/shopping etc) we put a piece of millet into his foraging toy inside the cage, and he's in like a flash! You'll could find something that your bird absolutely LOVES to encourage him to stay in the cage when you want him to, while keep persisting with his training so that hopefully he will just go of his own accord.
Hope some of this helps