Should Capacitors Have Resistance?

The resistance of an ideal capacitor is zero. The reactance of an ideal capacitor, and therefore its impedance, is negative for all frequency and capacitance values. The effective impedance (absolute value) of a capacitor is dependent on the frequency, and for ideal capacitors always decreases with frequency.

Do capacitors have resistance?

Capacitors, like batteries, have internal resistance, so their output voltage is not an emf unless current is zero. This is difficult to measure in practice so we refer to a capacitor's voltage rather than its emf. But the source of potential difference in a capacitor is fundamental and it is an emf.

Should capacitors have continuity?

Yes, but with caps that small it should not make a difference. Generally your meter measures continuity by applying a small current and then measuring the voltage drop. A short circuit will produce a very small voltage drop. If you connect a large capacitor that's discharged, this current will charge up the cap.