Use the Files→Save this dataset (.out)… menu. This stores the data in ASCII format together with the gnuplot script used to create the plot.
You can change the gnuplot options used to create the plots in the gnuplot settings dialog and save your settings in View→Profiles. For more advanced changes to the plots you can export the data and change the gnuplot script itself. Global changes can also be made in the gnuplot_preferences.py config file. Since Version 0.7.9 there is an additional option for 2D plots to change the appearance of each individual line under View→Change Plot Style. This also works for all datasets that you have combined for a multiplot.
Yes. You can fit some default functions as Gaussian, Voigt, linear regression etc. and their sums through the fit dialog accessible via the Data Treatment menu. Any other functions can be added using the IPython console.