This is the NMR for the molecule shown. I was told that carboxy group will be in the above 10ppm range. Does that tiny signal at about 12.9 ppm indicate the carboxy group though? I dont understand why the methoxy would show large integration while carboxy barely shows. Also my general question is, how much absorption is necessary for it to indicate some group present? Is there "noise" on NMR spectra or does every signal indicate a group?
Indeed, the very small signal at 12.9 ppm is due to the carboxyl group. The reason why it's so low is because, as explained here, at room temperature the proton exchange makes it nearly undetectable. At higher temperatures, you would probably not even see that spike.
The methoxy group, on the other hand, is not affected by such effects.
There will always be a tiny amount of "noise" on experimental NMR spectra, because a solution always contains unwanted elements; but usually anything that looks like a well defined spike is a significant signal.
Don't mistake peak intensity for peak area.
The (somewhat oversimplified) rule in NMR is that the area under the peak is proportional to the number of equivalent hydrogens, not the peak height.
If you look carefully at your spectrum, the methyl singlet is very very narrow - this will lead to a very tall peak (in addition to the fact that it's a singlet). Notice that it's considerably more than 3 times the height of the aromatic protons. The splitting of these protons into multiplets (difficult to see in your image, but you can deduce what they should be) gives them a shorter peak height than they would have if they were also singlets. The carboxylic acid proton is a singlet, but it's very broad and this leads to a dramatically lower peak height (you can see this in your image if you look carefully - but from experience I know that you often have to integrate over several ppm to get the integral to be even close to 1.
As @Hippalectryon mentions, this broadening is due to the exchangeable nature of this proton - how broad it is (and therefore, how tall it is) will depend strongly on temperature, concentration, the pH of your solution, how much water is in your sample, etc.
While probably not the case here, you can also get differences in peak areas when one proton has a significantly different relaxation time than other protons (due to motion, nearby quadrupolar nuclei, etc.) this will lead to differences in peak area even if they have the same linewidth assuming their T2* values are the same.