Air travel is fairly safe for a pregnant woman but more caution is expected in the first trimester which is the tender phase. Depending on the duration of the flight and possibly the route and health hazards in destination that may require immunization, the first trimester travel is not advised
Travel for a healthy, pregnant woman is generally fine. It is recommended though that you travel during your second trimester because you'll most likely feel better by then and the risks of miscarriage are low. It's always best to check with your practitioner before you fly to make sure you'll be alright.
Absolutely, unless other wise directed by your OB/GYN. I found it handy and comforting to have my doctor and other contact numbers handy in case or concern.