1) A lambda is an object without an identify and should not be used as a regular object.
This is not true. A lambda is a Java object, and it does have an identity1. The problem is that the lifecycle is deliberately left specified to allow Java compilers freedom to optimize the code that evaluates them. Therefore, you cannot know when new lambda objects are going to be created, or existing ones are going to be reused.
JLS 15.27.4 says:
"At run time, evaluation of a lambda expression is similar to evaluation of a class instance creation expression, insofar as normal completion produces a reference to an object. Evaluation of a lambda expression is distinct from execution of the lambda body.
Either a new instance of a class with the properties below is allocated and initialized, or an existing instance of a class with the properties below is referenced. ... "
2) A lambda expression should not be calling methods from Object class like toString, equals, hashCode etc.
As written, that is also not true. A lambda expression may call those methods. However, it is not advisable to rely on those methods to have any specific behavior when you call them on a lambda object
The JLS states:
"The class ... may override methods of the Object class."
In other words, it may ... or may not ... override them. If you rely a particular behavior of these methods, your application is (in theory) non-portable.
Furthermore, since the instantiation semantics are also unspecified, the behavior of Object::equals and Object::hashCode are uncertain.
Finally, it is unspecified whether lambdas are clonable.
1 - Sure, a lambda doesn't have a name: it is anonymous. But name and identity are different concepts.