I think that in each part (1,2,3) of this exercise the conditional component is not considered.
For example in the first, you considered p(x(t+1)) = p(x(t+1) | y1, ... , yt), but the knowledge of y1, ... , yt influences the density of x(t+1) since x(t+1) depends on y(t) (y(t) influences x(t) which in turn influences x(t+1)) and so on...
I think that you should use the formula of conditional Gaussian to compute p(x(t+1) | y1, ... , yt) since both x(t+1) and y1, ... , yt are Gaussian multivariate densities.
I think that in each part (1,2,3) of this exercise the conditional component is not considered.
For example in the first, you considered p(x(t+1)) = p(x(t+1) | y1, ... , yt), but the knowledge of y1, ... , yt influences the density of x(t+1) since x(t+1) depends on y(t) (y(t) influences x(t) which in turn influences x(t+1)) and so on...
I think that you should use the formula of conditional Gaussian to compute p(x(t+1) | y1, ... , yt) since both x(t+1) and y1, ... , yt are Gaussian multivariate densities.