Does this NMR show the carboxy group? 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.