Yes. It is good for you to continue to exercise while pregnant. Any exercise that doesn't involve a high risk of falling or getting hit in the belly is fine. With any exercise, you need to drink water throughout. You also have to be willing to back off if you start feeling headaches, nausea or dizziness that indicate your body thinks you're overdoing it.
Depends. Talk with your ob. It depends on the type of activities you would be doing in gym and on any health problems and any problems with the pregnancy.