If a person with a lazy eye donates his eye, will it stay lazy in the new person too?

No. Can't do full eye donation yet. Lazy eye relates to the brain input/connection to the eye, so the corneal can be transplanted without lazy eye issues.
No. In amblyopia, generally the eye is structurally sound with the brain's visual centers being affected. Generally only the corneas are considered for transplantation and if the corneal shape is acceptable, there should be no problems.