The statement is not necessarily saying that animals do not feel physical pain as humans do. The word ‘feel’ can be used for emotional pain too. Therefore, the statement can interpreted as animals do not feel emotional pain in the same way as us.
Now, we cannot determine how much pain an animal is suffering from. Pain is a sensation that only an individual can feel. (Therefore it is difficult to answer the second question).
We can surely understand if an animal is feeling pain, both emotional or physical. It is often observed that a dog becomes depressed when they do not see their owners for a long time. They tend to eat less and can often be seen sitting in one area waiting for their owners to return. We can therefore argue that some animals do feel emotional pain. Humans also feel a similar kind of emotional pain when we lose someone really close to us.
If physical pain is inflicted on an animal then we notice that they feel pain if they try to escape or retaliate. These behaviours can hence help to argue against the statement.
It is impractical and unreasonable to think that all animals have the same nervous systems. Therefore, others may argue that some animals may not feel pain since they have a different brain physiology. They may not have the part of the brain that we have that senses pain.