Thanks for the tag, @salmonmac.
@VWfarm, I was excited when I saw your first picture … it could almost be a double to my 2nd wheel.
Answers to your questions:
1- Your wheel is mostly an Irish style Saxony. (There is no brand … mine had an HG-7 carved on the end of the table). Not all of the pieces are original, including the drive wheel,the flyer, legs, wheel uprights … (you can tell because the turning on those do not match the more intricate turnings of the maiden uprights or distaff).
2- Your drive band keeps flying off because the drive wheel and the flyer whorl & the bobbin whorl do not line up. Make sure those two whorls are not glued together!! They need to be independent of each other. The flyer whorl should loosen by turning it to the right (the normal “rightey tightey, lefty loosey” doesn’t apply … it’s the opposite).