Can a child around the age of 6yrs old, get sick from handling animals like hamsters, geckos, etc...?

Yes: Simple hand washing with soap and water after handling will give ample protection from most infections but reptiles are known to carry salmonella, and younger kids will stick their fingers in their mouths before you can wash them for them.

Yes: Hamsters can directly or indirectly transmit Rabies, Tularemia, Salmonellosis, Lymphocytic choriomeningitis virus, Dermatophytosis (ringworm, dermatomycosis), Rodentolepiasis and Hymenolepiasis (tapeworms).