I'm not a MD, you need perhaps to have your GP do allergy screening IF you are indeed allergic to roaches (which I doubt as roaches are very fastidious and clean critters, they have to be survive and thrive in filthy situations).
Is more likely it's you are allergic to their fras or the molds that form on their food if it's not replaced before it forms molds.
You could use thin latex (or other material if you have latex allergy) gloves to handle them and their equipment. I think you'd have to be kicking up a LOT of dirt/frass to be concerned about respiratory allergy, but if you are, you could use a painters mask.
No, not only from handling. It can be from contact or the breathing in of particulates that causes the allergic response. But usually your symptoms will develop over time and become worse and worse, so you'll know as time goes on. It would be very unlikely to instantly develop anaphylaxis from your first exposure to them but it is possible. Gail who is a member here almost had to go to the er after being exposed to dubias. You might want to reach out to her, or ask her to post here for us.