I would choose the model or method for which I cannot find evidence of any pattern or structure that could be incorporated in the model.
Answer to your second question: Regardless of the model and method, the presence of some structure in the residuals indicates that there is some room for improvement of the model. Roughly speaking, the residuals are the part that the model cannot explain. If we can find a pattern for the dynamics of the data that remains unexplained in the residuals, then we can improve the model.
Back to model selection: If we stick to the information provided by the Portmanteau test, I would choose the model for which, given a significance level chosen beforehand, the null of independence is not rejected.
However, it is recommended to gather more evidence and information. As noticed in one of the comments above, failing to reject the null does not imply that we must accept the null hypothesis; there may be patterns in the alternative hypothesis that are not detected
or even considered by the test (e.g. heteroscedasticity).
In order to make a more knowledgeable decision, I suggest the following diagnostic tools (there are many other options which you have probably already considered): a plot of the autocorrelations and partial autocorrelations, a range-mean plot to check the homoscedasticity of residuals or a test for heteroscedasticy and a runs test to test the randomness of the sign of the residuals.