Crystal, or quartz crystal. Crystal resonator is not at all incorrect but they have been called just crystals for over 50 years to no harm...
Ceramic resonator; maybe just resonator, but certainly not crystal or oscillator.
Basically yes. (The middle terminal will be ground). However there are also ceramic filters with this configuration, so be careful and double check the datasheet. (Especially if they are marked with common radio IF frequencies, 455 kHz or 10.7 MHz!)
I don't know; most likely no.2 in different packaging, I have never seen quartz crystals packaged this way.
No, though connected to the right pins on an IC it might be. A crystal oscillator is either:
a. something like package (1) with 4 pins; it takes 3.3V or 5V in, and puts out a clean signal (making it dead easy to use), or
b. a circuit involving these components, possibly a resistor, and some kind of amplifier (inside the IC, or a separate transistor) or a misused logic gate, to drive the crystal.
In the latter case you can usually substitute a ceramic resonator, but you may have to change the capacitors and a gain setting resistor (or drive limiting resistor) to make it work at all, or work reliably.