We introduce and compute the generalized disconnection exponents $η_κ(β)$ which depend on $κ\in(0,4]$ and another real parameter $β$, extending the Brownian disconnection exponents (corresponding to $κ=8/3$) computed by Lawler, Schramm and Werner 2001 (conjectured by Duplantier and Kwon 1988). For $κ\in(8/3,4]$, the generalized disconnection exponents have a physical interpretation in terms of planar Brownian loop-soups with intensity $c\in (0,1]$, which allows us to obtain the first prediction of the dimension of multiple points on the cluster boundaries of these loop-soups. In particular, according to our prediction, the dimension of double points on the cluster boundaries is strictly positive for $c\in(0,1)$ and equal to zero for the critical intensity $c=1$, leading to an interesting open question of whether such points exist for the critical loop-soup. Our definition of the exponents is based on a certain general version of radial restriction measures that we construct and study. As an important tool, we introduce a new family of radial SLEs depending on $κ$ and two additional parameters $μ, ν$, that we call radial hypergeometric SLEs. This is a natural but substantial extension of the family of radial SLE$_κ(ρ)s$.