I have had degus both degus and gerbils. Unlike gerbils, it seems that degus don't tolerate being held very well. I think what you are doing – letting them run around the bed and sit on you – is a perfect way for them to get used to you.
I don't handle my degus except when I need a closer look at them. When I give them time out of the cage, they will come and sit on me but I don't generally hold them. And I catch them just the way you do – by waiting until they run into a tube.
How hard is the bite? My degus nibble a bit – hard enough to make me draw my hand away – but I don't think it is meant as a bite to scare me off. Their grooming can be rough.
I don't think covering the cage is a good idea. Is there any way to keep the cat away from the cage? It is risky letting the cat near them when you're not there to supervise and he might indeed be stressing them out.
I think the "wheep" you heard may have been an alert call. When mine do this, I just sit quiet as if I am part of their group. Within a minute, they're all usually back to doing their thing. I usually can't see what spooked them; however, they are very sensitive and even the sound of a car parking in our driveway puts them on alert.
Hope this helps.