There is no such bound. Let $u,v,w$ be orthonormal vectors. The matrices $A=[u,u+\varepsilon v]$ and $B=[u,u+\varepsilon w]$ have Q-factors $Q_A=[u,v]$ and $Q_B = [u,w]$ respectively, so $\|Q_A-Q_B\|$ is constant, but $\|A-B\|$ can be made arbitrarily small. You need to involve the condition number of $A$ and $B$ somehow.

There is a bound involving $\kappa(R)$; check Section 19.9 of Higham's book Accuracy and Stability of Numerical Algorithms.

There is no such bound. Let $u,v,w$ be orthonormal vectors. The matrices $A=[u,u+\varepsilon v]$ and $B=[u,u+\varepsilon w]$ have Q-factors $Q_A=[u,v]$ and $Q_B = [u,w]$ respectively, so $\|Q_A-Q_B\|$ is constant, but $\|A-B\|$ can be made arbitrarily small. You need to involve the condition number of $A$ and $B$ somehow.

There is a bound involving $\kappa(A)$; check Section 19.9 of Higham's book Accuracy and Stability of Numerical Algorithms.