- Are the residuals from a nonlinear regression model supposed to be randomly distributed too (as in linear regression)?
Yes. The error term is the same and the residuals represent estimated errors. The fitting of the model will induce some structure among the residuals, and that won't be identical in form for a nonlinear model as it is for a linear model but taking a Taylor approximation to first order, the nonlinear model is approximated by a linear model, so as long as a first order Taylor approximation is reasonable, even that structure will tend to be similar to that of a linear model near the optimum.
2. I am comparing two nonlinear regression models (non-nested). What model performance indicators can I use for the purpose? My understanding was to compare RSS, RSE, residual plots, autocorrelation and residual normality plots.
You can compare two different models fitted to the same data using those things, but what you should look at depends on what kind of performance matters for your present purpose.
Note that when you compare RSS, more parameters will tend to make the RSS smaller; you can't really compare them for models with different degrees of freedom. Similarly with MSE or RMSE. (You might better compare AIC or BIC instead, but I wouldn't use them as the basis for model selection - you need to properly account for out of sample error or you'll end up with the usual problems from doing model selection using the same data you measure model performance on)
3. The models have different Y axis.
If the response variable isn't the same, you can't really compare the models.
So I tried standardizing variables and then re-estimated RSS, RSE. Is that the right approach?
The right approach to achieve what exactly?