Hello and welcome to the forum.
A DMM (multimeter) in ohms or continuity mode tries to find the resistance between its leads by using Ohm's Law (V=IR).
Some of them do this by acting as a voltage source and measuring current, while others work by acting as a current source and measuring voltage. The voltage or current they source can be AC or DC. With all the possible combinations, it's easy to see how the results of measuring in-circuit are hard to predict. That being said, the continuity mode is normally set up to make it possible to find wire connections in a circuit. To do this, the meter should source a low voltage (less than a volt) and measure current. By only sourcing a low voltage, it avoids turning on semiconductors like transistors or diodes. There is still the matter of DC or AC: most meters use a low DC voltage but both are possible.
The continuity mode is intended for finding wired connections. Is there a trace between Pin 9 of U13 and the anode of D12? It can answer these kinds of questions confidently. It is not intended for measuring components either in or out of circuit. First, because there is no parameter of a component that is measured by a beep. Second, because different measurement setups are required to find the values of components like capacitors. Thirdly, most capacitors are not easily tested in-circuit at all, since they are most often used to bypass the power supply near each chip, which puts them electrically in parallel with each other.
What is the resistance of a capacitor? The question is badly formed, because it's asking to make a measurement of the wrong type of component. When a DMM in resistance mode is put on a capacitor, it can't find the resistance because the relationship between V and I doesn't act like a constant resistance. You can see anything from 0, to rapidly changing values, to a reasonably steady (yet wrong) value. If the capacitance is quite large (or there are many caps in parallel), it is rather likely to be 0, since it behaves closer to an ideal capacitor. Remember, an ideal capacitor has no series resistance at all!
The ceramic capacitors you have circled are not commonly subject to failure. It is possible for this type of component to fail if subjected to physical impact or excessive board flex, but not from shorting a 12V supply. My recollection is that the fans respond to temperature, so the driving circuit may have a voltage or frequency coefficent of temperature.
Finally, I would have sensible expectations about the probability of a successful repair. Any kind of learning process begins with simple objects of study and progresses to more advanced topics. When you have developed the requisite knowledge and experience, it becomes possible to conquer complicated problems like repairing highly-miniaturized circuits. To minimize disappointment you should write off the xbox now, since the approach you're taking is not working (the orange wire in the picture above reminds me of cat playing the keyboard).