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For a standard Op Amp as shown in the diagram, the 7microvolt supply is replaced with a 6.743 microvolt supply. Assuming the power supply to the Op Amp is plus/ minus 20V, what is the output voltage in volts?

Very silly question. The TL081 has an input offset of +/-3mV, so will swing to either rail for both conditions. How near it gets to the rail will be indeterminate.

This a very strange way to use an op amp which I have never seen before--- and I've had 30 or 40 years using them in industry. You can put a voltage on the non-inverting input to raise or lower the output but the inverting input works much better with a microamp current source. Microamps in gives negative output volts and conversely if you suck microamps out you get positive output. And you normally use a feedback resistor output to inverting input to get some control of the output voltage level . See the circuit for a more typical use.

Correct, but you forgot to allow for input bias current and offset (negligible in TL081) and input offset voltage (+/-3mV in TL081).

You're right but gufinsburg is obviously pretty new to the game and details like offsets and biases, at this point, would confuse the issue. The time to get into that is when fundamentals like Gain = Rf / Rin are solidly implanted.

Normally Ib and Vos are small but in this example Vos dominates over the teeny signals and completely swamps them.