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First of all congratulation on seeing the light, geese are the best, the more time you spend with them the friendlier they will get.
Duck mix is not the best for them, they do not eat meat in any form, where as ducks do, so you are better to buy a cheap horse mix, it will be better for them.
If you have a girl, yes they lay eggs, if you have two girls you will get lots of eggs, if you have a pair, you may get goslings and of course if you have two boy, sorry no eggs

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I would be best if you could lock them in a yard or pen for a week or so, they they now this is home. Whilst locked up, feed them some lettuce, cabbage leaves, etc whilst calling them to bed. They will learn that the calling means so good food. This will make it a lot easier in the future if you want to lock them up. Which you should do at night. In the yard have an area they can make a nest in, an old dog kennel or large wooden box, etc.
Introducing them to others is easy, just let it happen, if you have a rooster, he may want to dominate them, but this fight will only last a few seconds.
If the cat does not bother the chook, then it will not bother the cat, if the geese does not like the cat, the geese will tell it very quickly.
If you look in the Waterfowl Section, they are many topics about geese and ducks, have a read and if you have any question ask, we are here to help.
If you have a digital camera, we would love to see a picture of your new friends.