Is this circuit an FDNR?
The GIC based FDNR is one of the classical blocks for realizing active filters.
If you are using Antoniou`s GIC structure you will have no stability problems.

Regarding your last figure: An FDNR NEVER can replace a filter. It works - like a resistor - as a grounded one-port and must be combined with other passive parts (using BRUTON`s impedance transformation).  

UPDATE: The block as shown in the second figure (two opamps) is an ideal diffentiator because the feedback capacitor has no influence (feedback voltage is shorted due to the voltage source connected at the feedback node.

If you combine this circuit with a resistor (last figure), you have one of the classical realizations of a second-order bandpass (output at the 2nd opamp) or a second-order lowpass (output at the 1st opamp).

There is no relationship to any FDNR structure.