I agree with Simeon here that this is more a stat question.

First of all, the 2q score of -0.4 indicates that there is a negative correlation between summer and winter (i.e. the higher the summer values the lower the winer values). This allows for more predictivity so I wouldn't say the weakest.

1. These ACF and PACF indicate autocorrelations which are not significant (the significance interval is in blue). Maybe the agregation in quarters caused some loss of information here.

2. There is a lot of info on how to use ACF and PAF graphs in order to build ARMA models. These slides are a good overview of the associated Box Jenkins methodology and are a quick read: http://www.colorado.edu/geography/class_homepages/geog_4023_s11/Lecture16_TS3.pdf

I agree with Simeon here that this is more a stat question.

First of all, the 2q score of -0.4 indicates that there is a negative correlation between summer and winter (i.e. the higher the summer values the lower the winer values). This allows for more predictivity so I wouldn't say the weakest.

1. These ACF and PACF indicate autocorrelations which are not significant (the significance interval is in blue). Maybe the agregation in quarters caused some loss of information here.

2. There is a lot of info on how to use ACF and PAF graphs in order to build ARMA models. These slides are a good overview of the associated Box Jenkins methodology and are a quick read: http://www.colorado.edu/geography/class_homepages/geog_4023_s11/Lecture16_TS3.pdf

EDIT 1: Before going into the ARMA model, you should first differenciate your serie in order to make it stationary, i.e. with constant mean. Then you should be able to detect safely the periodicities of the time serie and try to build a model. If differenciating once is not enough to remove tendancy, do it again. You should really have a look at the Box-Jenkins methodology, which is really a standard for this kind of problem.

A small point: The test you are doing after building the model are don to test if you have a normally distributed error (or white noise) in your model. If so this means that you have discovered most of the information cointained in your signal.