Electronic – Will a resistor fixed just in front of an op amp change its gain

You will rarely see a circuit with just one resistor as you show it; usually there will be another resistor (or equivalent source resistance) of the same value on the noninverting input, too.

Most (nonideal) opamps have a finite input resistance, and this means that a tiny current flows into or out of the input terminals. This current is called "input bias current", and it varies with the voltage at the inputs. Since most opamp circuits use negative feedback to keep the two inputs at the same voltage, this means that for any given voltage, the current through both inputs will be the same.

The current through each input flows through whatever resistance is connected to that input, and this introduces a voltage shift at the input. If the resistance at the two inputs is different, this voltage shift will be different, too, and the difference between those two shifts will appear as an additional input offset error in the operation of the circuit.

For this reason, an effort is made in all opamp circuits to make sure that the resistances connected to the two inputs are the same, eliminating this additional source of error. Even in a unity-gain buffer, if the source resistance is 100Ω, a 100-Ω resistor will be used in the feedback path.

Positive feedback and negative feedback can be combined to create oscillation (astability). Aside from that, I think you're correct. In the case of the integrator, there's never a step change in the output, so there's never a discontinuity in the capacitor voltage. The only limit is the output voltage range of the op amp.

Assuming an infinite gain with negative feedback implies that the voltages at the two inputs will be equal. Assuming an infinite gain with positive feedback implies that the output will be saturated (unless V+ exactly equals V-).

Even in a real op amp, there's very little current into the input pins (microamps at most). Likewise, the open-loop gain is usually very large at low frequencies (on the order of 10^5 or 10^6 V/V). So the ideal op amp rules are often a pretty good approximation. But all real op amps are limited by their supply voltages, so sometimes that limit is included in an ideal model as well.