How to calculate voltage on a multi-powered DC circuit where all ends go to ground

The question which noise sources need to be taken into consideration depends on how severe they are. Your question indicates that you are interested in noise generated at the op amp and not noise generated by interference from neighboring circuits (internal/external noise).

In order to make things comparable, all noise is referred to the op amp's input (RTI). In theory, I guess any point in your circuit might work as long as you refer all noise sources to that point, but it is common practice to act as if all noise sources were directly at the input pins. Sources include noise in the resistors, noise generated by current flowing into the op amp's input pins and noise that may be considered as a voltage between the inputs pins.

There is a very good discussion at this Q&A-style source and also in this nice article from 1969 (!), both authored by Analog Devices' staff.

Without re-typing everything in these sources, here are some rules of thumb:

Noise in the resistors becomes bad when the resistor values are high (some 100k or some 1M) and when the circuits are designed for high bandwidth since the noise is proportional to \$ \sqrt{4k \cdot T \cdot B \cdot R}.\$

You can try to minimize R, you can try to limit the bandwidth B if possible, you can put the circuit in liquid nitrogen (low temperature T), but you can't go for a low Boltzmann constant, because Boltzmann is dead (quote stolen at Analog Devices).

Current noise, i.e. noise generated by current flowing into the op amp inputs, will be converted to a noise voltage by the resistors around the input (\$R_f\$, \$R_g\$) and amplified by the circuit's gain. This is one of the reasons why one prefers op amps with very low input currents especially for high-ohmic circuits.

Voltage noise results from a real op amp's inability to completely null the voltage between the input pins.

All noise sources can be combined as the square root of the sum of their squares since they are independent of each other, which will work only if all sources are RTI.

My favorite educator, Bill Beaty, often rants at the many misconceptions that all too many people have been infected with. One of the many common misconceptions involves batteries.

"Frequently-Asked Electricity Questions": "THE LIQUID BETWEEN A BATTERY'S PLATES IS A GOOD CONDUCTOR. SO WHY DOESN'T IT SHORT OUT THE BATTERY?"

"Why is electricity so hard to understand?" "...mistaken belief that no charge flows through batteries. ... This leads to the traditional incorrect flashlight-current explanation (current comes out of battery, flows...etc.) It also leads to the misconception that batteries SUPPLY CHARGE, and have a storage place for "used" charge. This might make sense if we believe that there's no path for charge through the battery. But it's wrong, because there is a path, a path provided by flowing charged atoms. Charge must flow around and around a circuit, passing THROUGH the battery over and over."

"But how SHOULD we teach kids about 'electricity'?" "A battery is a chemically-fueled charge pump. Like any other pump, a battery takes charges in through one connection and spits them out through the other. A battery is not a source of the "stuff" being pumped. When a battery runs down, it's because its chemical fuel is exhausted, not because any charges have been lost. ... When you "recharge" a battery, you are pumping charges through it backwards, which reverses the chemical reactions and converts the waste products back again into chemical fuel."

'Which way does the "electricity" really flow?' "When you connect a lightbulb to a battery, you form a complete circuit, and the path of the flowing charge is through the inside of the battery, as well as through the light bulb filament. Battery electrolyte is very conductive."