You are correct in saying a particular event is not associated with a particular time.
This does not mean that time is derived, though. It's just different for observers travelling at different speeds.
An event is just a point in spacetime. Usually, we care about interesting events, like a lightening strike, or an explosion, or a train passing a flare or whatever. The catch is that observers travelling at different speeds will measure events to be in different places in spacetime: either squished closer together or stretched apart. However, events will always be in the same order (both in time and space), and events happen no matter who's looking.
You could see spacetime as a stretchy (4-dimensional!) rubber sheet, and events as dots on the sheet. You can pull at the sheet to change how far apart the dots are, but you can't swap or erase dots. Here, the pulling is analogous to changing reference frame. The behaviour of this sheet at constant velocities is described in Special Relativity.
As for the last point: they're all space-temporal! They're all derived in the sense that they differ between observers (this is called non-invariant), as I said about time above.
I hope this has cleared some things up.