1: As I previously said an op amp is just an amplifier block with high gain and differential inputs. It's not of much use without additional external components to configure it to do what you want. It can be configured to be an inverting amp, non-inverting amp, summing amp, integrator, differentiator, etc., with a defined gain as determined by these additional components.
2: There are many other kinds of amps: audio amps, RF amps, microwave amps, etc. Op amps are just very common since they can be configured to do so many things.
3: Yes. An op amp can be configured to amplify small input currents to generate larger currents or an output voltage proportional to the input current (trans-impedance amp).
4: There is both positive and negative feedback. Positive feedback generally leads to amplifier oscillations or latchup so is not generally intentionally used. Negative feedback takes a sample of the signal from the output and applies it to the amplifier input such that it reduces (stabilizes) the amplifier gain and reduces distortion.
A simple example is an op amp configured as an inverting amplifier (below) with the (+) input grounded, input resistor (Rin) connected to the (-) input, and a negative feedback resistor (Rf) connected from the op amp output to the same (-) input. With that configuration the high gain of the op amp, will cause the op amp to always try to keep the (-) input very close to the same voltage as the (+) input (in this case the same potential. as ground). This means the current trough Rf must equal the negative of the current though Rin.
Think about why this is so. Suppose that the current from the output is slightly higher through Rf into the (-) junction then that through the input Rin. This will cause the (-) voltage to rise. The amp gain then causes the output voltage to drop and reducing the current through Rf until it matches the input current.
Since the two currents are equal then Vin/Rin = -Vout/Rf. Cross multiplying gives the value for the gain as Vout/Vin = -Rf/Rin (the minus means the gain is negative, it's an inverting amp). Notice that the closed-loop gain is determined by the value of Rf and Rin, not the op amp gain which is normally several magnitudes higher.