Almost every gene (the exception being sexlinked genes) comes in pairs - one from mom and one from dad.
Blue is simple dominant, so if a chicken has a single copy of the blue egg gene the eggs will be blue. If they have two copies the eggs will be blue.
Like this...
I'm sorry you're dealing with this. What you describe doesn't sound like a very good quality of life. Remember, chickens try to act as normal as possible even when they're in pain... So if he can't walk properly there's probably something deeply wrong and/or causing him pain.
Only you can decide...
But "for the animals" means so many things do different people. Full disclosure - I'm NOT vegan. But I know a lot of folks that are vegan, vegetarian, pescatarian, etc. Some I'm good pals with, some I'm not. The ones I'm friends with tend to be very low key about it.
I suspect the more...
This depends on who you talk to.
Some vegans actually do believe that an animal in captivity is better off not having been born.
Some vegans believe that doing anything to take and use the products of animals is wrong, even if they are natural and excess. I guess here, you're using these eggs...