This varies enormously, from societies with fully casual gender mixing to those where unrelated men and women maintain careful public distance, so calibrate to your destination early by watching how locals behave. A few defaults, though, keep you respectful across the whole spectrum.
On contact: in conservative Muslim, Orthodox Jewish, and some traditional communities, physical contact between unrelated men and women, including handshakes, may be avoided; never extend your hand first across genders there, and if no hand comes, a smile with a slight nod is perfectly gracious. In liberal settings, normal professional and social mixing applies and excessive caution would itself seem strange. Casual touches on the arm or shoulder across genders are best dropped anywhere conservative.
On conversation and eye contact: friendly, purposeful interaction is fine nearly everywhere, but in traditional societies, prolonged one-on-one attention, personal compliments on appearance, or sustained eye contact across genders can be read as romantic interest, and the social cost may fall on the local person more than on you, which is the real reason for restraint. Some settings, including certain religious sites, transport sections, and traditional gatherings, seat genders separately; follow signage and example without comment. In private homes, let hosts choreograph introductions and seating. Overall, err formal and reserved until local friends show you where the lines actually sit.