Yes, the eggs do not have to be fertilized.
Yes a bird can lay eggs with out a mate. You have to be very careful because you do not want this to keep on happening due to it takes up lots of her calcium with each egg that she lays. My own bird did not lay eggs three in all until a couple of weeks ago. I am now making sure to replace her calcium with lots of veggies that have lots of calcium in them. My bird bonded with me and to her I became her mate so now I have do do things to stop her to get her interested in other things besides me so she now even has a play stand lot's of toys and I go out more. Be careful because of you do not want her to get any stuck inside of her which as the last person told you it can result in death. My African Grey went to the vet and had her last egg there where they even gave her shots to stop the laying.
Yes they do. Birds do not have to have a mate in order to lay eggs. However,they do need a mate to lay fertile eggs, and have chicks. On many farms, hens lay eggs without roosters being present, and if a rooster is present, he is kept away from the hens.