The only things that can be shared between the two species is rabies and worms and other parasites (like ringworm). Also, sneezing is not a sign of doggie colds. Coughing is. Sneezing usually indicates something irritating the upper respiratory tract (something he inhaled). Usually lethargy is not concurrent. See your vet if you are concerned.
Best wishes.
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Cheers,
Jessica
10.15 Ball Pythons; 8.9 Corn Snakes; 0.0.1 Green Tree Python
2.2 Jungle Carpet Pythons
1.0 Cay Caulker
1.0 Nic, 0.1 Tpos Nic
1.0 Hypo Het Albino BCI, 0.1 Albino BCI
0.0.1 BCC (almost 20 yrs old)
3.0 Crazy Dogs and 2.0 cats
Some Tropical Fish
...........and growing!