Just to add a little bit more, both ".log" and ".trees" are log files which contain the sampled states from the MCMC. For instance, if you run 10x6 generations and you sample every 1000 states, both your ".log" and ".trees" files will contain 10,000 samples (or states). The difference between both files lies in that one (.log) is in a table format and includes the sampled states of each and every parameter in your model (or suit of models, e.g. substitution model, molecular clock model, transition rates from discrete CTMC, calibrations, etc.), and their associated likelihood scores, including the prior and posterior. The ".trees" file includes essentially the same information for every sampled state, but in the form of annotated ultrametric trees. Regularly, you would use Tracer to visualize your ".log" file, and determine if your analysis reached convergence (e.g. by comparing multiple runs) and if there was sufficient mixing in the model parameters throughout the MCMC (by looking at the ESS values). If your analysis passed the test, regularly you would summarize the information in your posterior trees (.trees) into a single, most informative, maximum-clade-credibility (MCC) tree. Then you can visualize your results by opening your MCC tree in FigTree, for instance, to look at the topology, the divergence times, the level of rate variation among lineages, or the evolution of a discrete or continuous trait (and for each, their level of uncertainty, e.g. 95% HPD intervals).
On your second question, although BEAST can estimate (for tree building and parametric purposes) birth and death rates (i.e. speciation and extinction rates), normally you would do diversification rate analyses with other, more specialized, software, using your MCC tree or maybe a subsample of your posterior distribution of trees. Some of these programs or packages (e.g. BAMM, DDD, PBD, Diversitree, Geiger, phytools) can perform diversification rate analyses, but vary in the model implementation.
Cheers,
Santiago