Is there a correlation between asthma and eczema?

Yes. Both eczema and asthma can occur due to an allergic etiology. So, the 2 conditions can have the same etiology in a given patient.
Usually. Asthma, eczema, and allergies are all atopic diseases, where that part of the body is overrreacting to something that other people don't react to. They can run in families but sometimes occur alone. There's no 'cure' but they can be controlled. Your doctor can work with you to find the best treatment regimen.
Yes. Both are part of atopic disease., usually people who have one have the other. Treatment is also similar.
Yes. Asthma and eczema are both allergic type disorders and many patients with one disorder will also have the other.