Estimate of mean after a time series approaches equilibrium

I am working with a time series for a quantity that undergoes some relaxation towards a mean value, after which it appears to exhibit strong noise and possibly some very weak drift. Here is a graph of the time series:

While I can eyeball a good estimate for this mean, I'd like a more objective method to determine the following:

1) Time to approach equilibrium

2) Equilibrium mean, and linear/nonlinear drift (if any)

3) Some sort of stationarity test of the noise

I have been playing around with some ARIMA tools in R, but I think that I'm out of my element here, and that I ought to get some more experienced advice.
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