while you were walking walking the dog ,you were careless an

是的
while （you were ）walking walking the dog ,you were careless and it got loose and was hit by a car
主语一致,可以把从句的主语省略
再问： 必须满足这两个条件吗？ 主从句主语一致 和 主语和动词是主动关系
再答： 对的