Yes, negative 1H NMR shifts do exist, for example the inner Nitrogen-bound Protons of porphyrins usually have negative chemical shifts (ranging from -1.5 - -4.5 ppm) due to the strong shielding effect of the macrocyclic aromatic ring current on Protons inside the macrocycle (as opposed to Protons outside the macrocycle ...

Can you have negative PPM?

NMR is based on quantum physics and it is not possible to get a negative concentration value.

Which carbon is most downfield?

The ¹³C-NMR signals for carbonyl carbons are generally the furthest downfield (170-220 ppm), due to both sp² hybridization and to the double bond to oxygen.