What is the methodology to analyze future climatic extremes using the results of climate models?
Your question is fairly broad, and slightly vague so I don't know if I will be able to answer it in one go. If you would like any clarification, let me know.

How are predictions made?

The basic process generally used for probabilistic forecasting of the global climate is something like:


Take one or more general circulation models (GCMs), and run multiple simulations over the projection period.


GCMs are models that attempt to replicate the physics of the climate system, including land and ocean processes. They can not replicate the physics exactly, because:


they operate on scales of a few hundred meters at minimum, and so physical processes (which usually operate on much smaller scales) are scaled up and parameterised to work on the larger scale; and 
because for some of the physical processes (especially those that have to be parametrised to larger scales), there is not enough data to adequately fit them. When this happens, those model parameters are usually calibrated - a large number of runs are made, with different values for those parameters, and then the best parameter values are chosen.

Usually, you run the model for some period prior to the projection period, so that you can compare that to observations (spin-up period).
You need to provide input data to the model. Mostly, this just consists of solar forcing (changes in the sun's output), as well as expected greenhouse gas (GHG) emissions, and aerosol emissions (both human and volcanic).
Simulation output is in the same format as your observational data, and generally includes all of the important variables - temperature, precipitation, etc.

Put all of those simulations in an ensemble.


Often, simulations are "bias corrected", which usually means they are changed so that the model average matches the observation average over some period prior to the projection period - this might be the last 30-40 years. This is to allow for model drift (which is mostly chaotic, and can happen in any direction).
The distribution of the simulations in the ensemble is used to create probabilistic projections. For example, each simulation has a mean global annual temperature. You could make the assumption that the errors in the simulations are normally distributed (over the ensemble), and so make an estimate of that distribution, and use that to make probabilistic estimates. This is basically saying that if ~50% of the models are over a certain temperature, you expect that (assuming the models, the input data, and assumptions they are based on are reliable*) there's a 50% chance that the global annual temperature of the earth would be over that temperature. A similar process can be used for the maximum temperature, or any other variable, and any statistics, including extremes indices. None of these predictions is a single value - they are always probabilistic, although it's probably quite common for only the mean of that distribution to be reported.
How you combine the information from the simulations in the ensemble is somewhat up to you. The basic method is as described above, and uses minimal assumptions about the data. It's also common for researchers to use a weighting methodology to estimate the ensemble mean, for example performance-based weighting, where models that perform better over the period before the projection are weighted more highly, with the assumption that they will perform better over the projection period too. Or you could weight based on model independence (if you're using multiple models), by weighting models that are less like other models more highly, in order to minimise bias from particular types of models (e.g. Bishop and Abramowitz 2013). [I have a paper in review at the moment that investigates the efficacy of some of these methods, I'll link to it when it comes out]
Can this process predict changes in 1000-year events? Yes, potentially, even when the projection period is usually more like 100 years. This is because 1 in 1000-year events are actually defined inversely - that is, they are thought to have a 0.1% chance of occurring in any given year. If your ensemble projects over 100 years, but has 100 simulations, then there are 10,000 years in which that a 1 in 1000 might occur. If you see it occurring 20 times, or 4 times, then you might get suspicious that things are changing. You probably need more than that to get a statistically significant result, but there are ensembles out there that are really big (e.g. Stainforth et al, 2005, which had 2,578 simulations totalling over 100,000 simulated
years, and  I think some of the ClimatePrediction.net have tens of thousands of simulations). Even without those large ensembles, if you assume particular distributions of extreme events where probability is a simple function of extremeness (e.g. Log-normal distribution or similar), then you can probably say something about large extremes based on the change in behaviour of less extreme events. Unfortunately, I am not aware of any experiments that look at the likely changes to very extreme events.



* Model reliability is obviously a fairly active field of debate and research. Suckling and Smith (2012) indicates that GCM-based ensembles are better than basic statistical models, but probably not currently much better than some more advanced statistical models. However, that still means that they're at least on-par with the best options we have for forecasting, unless you're in to reading chicken entrails.

References


Bishop, C.H. &amp; Abramowitz, G., 2013. Climate model dependence and the replicate Earth paradigm. Climate Dynamics, 41(3-4), pp.885–900. Available at: http://link.springer.com/article/10.1007/s00382-012-1610-y [Accessed September 17, 2013].
Stainforth, D.A. et al., 2005. Uncertainty in predictions of the climate response to rising levels of greenhouse gases. Nature, 433(7024), pp.403–406. Available at: http://www.nature.com/nature/journal/v433/n7024/abs/nature03301.html.
Suckling, E.B. &amp; Smith, L.A., 2013. An Evaluation of Decadal Probability Forecasts from State-of-the-Art Climate Models*. Journal of Climate, 26(23), pp.9334–9347. Available at: http://journals.ametsoc.org/doi/abs/10.1175/JCLI-D-12-00485.1 [Accessed January 8, 2015].