Everything that was originally asked for can be done by windows already.
I know Windows does this natively, but I would love to be able to just select which icons would always be hidden, and show them again with a mouseclick. Anybody know?
Right clicking on the clock in the taskbar (or any other blank space in the system tray) and selecting "customize notifications" brings up a dialog similar to my first screenshot. You can also get there from the 'customize...' button on the taskbar properties dialog.
I've highlighted the three different choices, "always hide", "always show", and "hide when inactive". The last is the default. Icons that are always hidden will not be shown until you open the systray, by clicking the arrow button. When the systray is closed, active icons and those set as "always show" will be present.
In my use, I find an icon rarely needs to be specified as "always show", but several will be "always hide".
Of course this isn't perfect. It may not do everything you want. However, it will work well for many people -- once they find out about it.