The acf is the ratio of the covariance to the variance. If you have pulses/level shifts/seasonal pulses and/or local time trends (my guess is YES!) then both the covariance and the variance are affected. The net result is that there is a downward bias to the calculated acf. I suggest that you post your time series and I will put it under a microscope. The problem with the DW test is that it only tests for lag 1 autocorrelation and of course following my discussion above is seriously downwards biased when anomalies are present. Yes Virginia, there is no Santa Claus! In closing I believe that more advanced(correct) tools are required.
It is possible to get a general formula for stationary ARMA(p,q) autocovariance function. Suppose $X_t$ is a (zero mean) stationary solution of an ARMA(p,q) equation:
$$\phi(B)X_t=\theta(B)Z_t$$
Multiply this equation by $X_{t-h}$, $h>q$, take expectations and you will get
$$r(h)-\phi_1r(h-1)-...-\phi_pr(h-p)=0$$
This is a recursive equation, which has a general solution. If all the roots $\lambda_i$ of polynomial $\phi(z)=1-\phi_1z-...-\phi_pz^p$ are different,
$$r(h)=\sum_{i=1}^pC_i\lambda_i^{-h}$$
where $C_i$ are constants which can be derived from the initial conditions. Since $|\lambda_i|>1$ to ensure stationarity it is very clear why the autocorrelation function (which is autocovariance function scaled by a constant) is decaying rapidly (if $\lambda_i$ are not close to one).
I've covered the case of unique real roots of the polynomial $\phi(z)$, all other cases are covered in general theory, but formulas are a bit messier. Nevertheless the terms $\lambda^{-h}$ remain.
Answers to question 2 and 3 more or less follow from this formula. For $AR(1)$ process $r(h)=c\phi_1^h$ and when $\phi_1$ is close to one, i.e. close to non-stationarity, you get the behaviour you describle. The same goes for general formula, if the process is nearly unit-root one of the roots $\lambda_i$ is close to 1 and it dominates other terms, producing the slow decay.
Best Answer
The plot that you show seems very close to the typical ACF of the fundamental seasonal cycle in a monthly series. The periodicity of this cycle is annual, it is completed once every year. That could explain the 6-months between a peak and a trough in the ACF and the 12 months for the whole cycle peak-trough-peak. The following model generates such seasonal cycle:
$$ y_t = \sqrt{3} y_{t-1} - y_{t-2} + \epsilon_t \,, \quad \epsilon_t \sim NID(0, \sigma^2) \,. $$
In R, some data from this model can be generated as follows:
Looking just at the ACF, the presence of an annual cycle seems compelling. However, as @NickCox mentioned, you should look at other plots and sources of information (original data, periodogram, partial autocorrelation,...) in order to choose a model for the data or describe the features of the time series.
There may other relevant cycles in the data. One of your next steps in the analysis could be to check if the filter $1-\sqrt{3}L+L^2$ ($L$ is the lag operator such that $L^iy_t = y_{t-i}$) renders the data white noise or if there is some remaining pattern to be identified: