Yes - GIMP can do precisely what you are asking for. As it is a somewhat advanced demand, it is not that easy to find about it - but I think it the functionalities requested are well placed into the program.
The Key for what you are asking:i.e. "Is there a way to save a combination of these settings, so that tool, brush tip, size, color and opacity are preselected and I have a palette I can use to switch between them with a single click?"
is to open a dialog window called "Tool Presets" (open it at the windows->dockable dialogs->tool presets menu option)
There you can see some factory presets - of course these won't suit you.
To use it, you first configure one of your tools exactly how you like them to be: brush, brush size, opacity, FG/BG colors and so on, and once you are done, click on the "new preset" button (the second button on the bottom of the dialog - only the icons are shown) on this dialog - you are taken to another dialog: the "tools preset editor" - just type in a name (like "my blue pen") for your tool, and mark the appropriate options (for example: apply stored FG/BG for your examples above) - the tool configurations besides the selected resources like active color/brush/gradient don't need to be configured at this "tool presets editor" dialog - they are taken from the state your tool was when you pressed the New preset button on the previous dialog. (i.e. the parameters like opacity, size, painting mode, and so on)
There you are: it is automatically saved in your GIMP configuration personal folders, and available even if you close the program and open it again. Go back to the tool presets dialog: You will find out that no matter what is your tool and color configuration, by clicking on the "my blue pen" on the tool presets dialog to have your pen configurations restored.
Now, just repeat the steps above to configure your other tools - like the pencil and eraser.
And there is one extra feature as a bonus: another little known feature in GIMP is the ability to tag resources so that you can quickly find them.This works for gradients, palettes, brushes and tool presets: Still on the tools preset dialog type in a tag, on the text entry box at the bottom of the dialog, for the presets you have created - say "mytool". After doing that, just type teh same tag on the text entry on the top part of the dialog: voilá: only the tool presets you just configured are visible now. This gives you exactly "a palette I can use to switch between them with a single click" :-)
Here is the link for the relevant section on the user manual:
http://docs.gimp.org/en/gimp-dialogs-misc.html#gimp-presets-dialog
update: It is interesting to note that the configuration values are snapshoted at the moment one creates the new tool preset. There is no way to update that from within the program - the UI misleads you - it looks like you could update the tool options and press "save preset" on the dialog again, but this has no effect. Yes, this is a broken behavior on the UI an we (GIMP's project) need volunteers do redesign and code improvements on that if one would step up.
(Note that for advanced usage, the values can be updated using a text-editor, since GIMP's pre-set files are just text configuration files)
