kalman filter with time-varying noise?
1) It depends on what you call the standard Kalman filter -- I will call the equations in the picture below to be the "standard Kalman filter". You can easily derive an expression for the Kalman filter where the covariance matrices of the noise processes are time varying in terms of the covariances of the noise processes (and cross-covariance of the noise processes). 

Moon and Stirling's Mathematical Methods and Algorithms for Signal Processing derives the Kalman filter under $w(t)$ has mean $0$, covariance $E[w(t) w(s)^T] = Q_t \delta_{ts}$, $v(t)$ is also mean zero with covariance $E[v(t) v(s)^T] = R_t \delta_{ts}$ and $E[w(t) v(s)^T] = M_t \delta_{ts}$ (if you assume $M(t) = 0$, the equations simplify considerably). 

For the system model 
$ x_{t+1} = A_t x_t + w_t , y_t = C_t x_t + v_t$  [$w_t = w(t), v_t=v(t)$] with $M_t = 0$, the following are the resultant Kalman equations (from Moon &amp; Stirling) to get the predicted state estimate (13.45) and filtered state estimate (13.47):



These equations can be shown to be optimal using either a Bayesian approach (MAP with Gaussianity assumptions) or Linear MMSE approach for the given system model. 

2) Convergence of the Kalman filter is a bit of an annoying subject. You can read these notes for more details or check the book Stochastic Systems: Estimation, Identification and Adaptive Control by Pravin Variya and P.R. Kumar or one of the other texts on stochastic systems. Essentially, if the systems are controllable and reachable and yada yada, you can get some statements on convergence properties.