Yes. When ever you cross time zones your internal clock as well as the childs become affected. If you travel with a small infant it will not matter as much as infants sleep and eat on demand anyway. For older kids you might want to start changing their bedtime by about 15-20 minutes every couple of days to prepare them for the change. As soon as you get there, go to bed and eat meals at their time.
Yes. Babies, children, and adults all have an internal bodily rhythm that is based on our 24-hour day. So, everyone can have jet-lag.
Yes. You and your baby are susceptible to jet lag which is cause by rapid (plane) travel east or west. Based on personal experience, your child is likely to recover faster than you. Spend time outside in daylight, especially early in morning and evening to help reset the internal clock. Our internal clocks are set by sunlight and the body's natural sleep hormone, melatonin.
Yes. As any other traveler will experience an adjustment to a new time zone, so will a baby. Get your baby back into the routine you had established on your own home turf and she will acclimate as you do once in your new location.