I know a few spinners who have used their dogs fur- one even does it as a business and will knit you a teddy out of it
It helps to mix it with a well crimped sheeps wool but if you can make yourself a spindle, tis a lovely hand-project to wander around the woods chatting at campfires with
Goldings are the rolls royces with their rim weighting top whorl designs (much easier to stand and spin with in the the woods compared to the bottom whorls) or theres a simple instructable
here... I'm mid-spindle make atm as I lent mine to a bushbro who wanted to make them himself and I've never got 'em back!
Sky's the limit for design, just keep within the range of weights/ sizes on the goldings specs, lighter for fine wool (which is a LOT harder to spin evenly!), heavier for coarser, thicker wool... then you can make fingerless mitts easily with a rectangle of it knitted up
The problem with a bowstring is there's always some inherant elasticity in a spun fibre, particularly one made of many shorter fibres- it'd prob not have the strength as it slips along the fibre. You'd need a strong long (poss veg/ bark?) fibre in which case the dogfur would be purely aesthetic and likely to shed out of the string quickly.