Electronic – What are the differences between active bandpass filter topologies

A filter topology, whether it's a passive one like a pi-section or an active one like the Sallen-Key circuit, is just a way to produce some poles and zeros. Generally, you can tune the circuit values (resistances, capacitances, inductances) to move those poles and zeros around in the s-plane.

A filter design, like Butterworth or Chebychev, is a choice of pole and zero locations that gives a certain performance (Butterworth has maximally flat pass-band, and Chebychev filters minimize the error between the real filter and a boxcar filter, for example).

You can use whatever filter topology you like to implement whichever filter design you like. First determine the desired pole and zero locations from the filter design. Then tune your component values to achieve those locations. However, you might find that some topologies require unreasonably large or small component values for certain filter designs, or be excessively sensitive to errors in the component values, and that kind of issue might motivate you to use a different topology.

As to how they were invented, I suspect the guys who came up with them were simply very very clever.

There are several different active bandpass topologies available. The most common are Sallen-Key and Multi-Feedback. Show us your first attempt - so we can discuss on a realistic basis. If you are a newcomer in filter design I urgently recommend to use on of the filter design programs (online or downloadable). But note that you need a clear specification (filter order, mid-frequency, bandwidth resp. Q-value). And - if possible - use split supply only!

Selecting a suitable opamp should not be a problem in your case because the operational frequency range is sufficiently low (by the way: Don`t care about common-mode input voltage ranges). Most probably you got "strange" results (negative dB values) because the opamps were not powered correctly (single supply).