The gander ratio problem is why I bought sexed goslings only. Last year my geese wandered off. They sometimes wander into one neighbor's yard where we can see them and we call them back but this time they decided to walk through the woods (they never go into the woods) to another neighbors...
My geese are specifically for laying. Just like chickens. Except geese will lay only during their breeding season (usually sometimes an odd fall egg) which is around the end of January to June or so.
I use a stick of some sort as well. You might try training them to the stick while you are home so next time you are away the goose sitter will be able to use the familiar item and things might go easier.
I put them in the coop at night with the chickens. They are allowed to go into the secure run and most of them do. They will eat all the chicken food and drink all the chicken water so I have to replace that in the morning.
Aww, poor thing. Good care, green pasture, and fresh air should help some. Geese are not really amenable to being kept in cages. They need to be out on pasture. I don't know if anyone even knows what their true nutritional needs are.
You don't need more than two to start I think. You can always grow out more later. I am no expert and I am a coward. I meant to eat the boy geese and keep the girls (because I sell eggs) but so far I have not had the courage. I have no problem with chickens, but the geese are more personable...
My goslings are growing like weeds but they just don't eat that much of the crumbles. They prefer the pasture. Wild geese don't have access to crumbles anyway so I am not worried.
Just to throw in another reason people might want Africans, I have been doing really well with my Africans from McMurray Hatchery. I don't care what the size is or even what the look is or how close to show quality. I buy them for egg production and egg size and great foraging. I sell my eggs...