1. What do you want the noise to do? If you want to check whether you noiseless systems behaves the same way as the one with noise -- introduce zero mean white noise times small number $\epsilon$. Where to put (additive noise or parameter noise) it is your choice, a priori you can get different answers for different places.

2. Depends again on what do you want this time lag to represent. The most naive option is just to replace $x_2(t)$ by $x_2(t+constTimeLag)$ in the second system of equations (and the similar for $y_2$ in the first system). Of course you will not be able to use standard ode solvers because it is a systems of ODE with delay which are more difficult to solve (and needs heavy mathematics to be accurately described).

The question about units I did not understand -- you have a system without units at all.

1. What do you want the noise to do? If you want to check whether you noiseless systems behaves the same way as the one with noise -- introduce zero mean white noise times small number $\epsilon$. Where to put (additive noise or parameter noise) it is your choice, a priori you can get different answers for different places.

2. Depends again on what do you want this time lag to represent. The most naive option is just to replace $x_2(t)$ by $x_2(t+constTimeLag)$ in the second system of equations (and the similar for $y_2$ in the first system). Of course you will not be able to use standard ode solvers because it is a systems of ODE with delay which is more difficult to solve (and needs heavy mathematics to be accurately described).

The question about units I did not understand -- you have a system without units at all.