Will two op-amps connected to the same input affect each other?
To ensure the two circuits have the correct intended gain, a strong voltage source should feed node 1 and node 2. In other words the output impedance of the voltage source has to be very low or the amplifiers will not have the gain as expected.

Because of this, it's reasonable to say that any loading (due to adding the number of amplifiers in parallel) will have negligible effects.

If the voltage source is not considerably low in its impedance then neither op-amp circuit will work with the gain they are expected to deliver.

Regarding the effect of a feedback resistor, are you aware of what a "virtual" earth means and the spin-off it has on differential amplifiers?