To stay calm during sparring, approach sparring as an opportunity to learn rather than a performance. It helps to stay calm if you train properly, rest well, meditate, and talk to your sparring partner before a sparring session.
You can spar with training gloves. Both sparring gloves and training gloves protect your opponent but training gloves also give more protection to your hands. This means that sparring in training gloves provides less real-world training than sparring in sparring gloves, which are meant to mimic real fights as closely as possible.