If none of your immediate family members are allegric to general anaesthesia, does that mean you're safe to go under?

No. There are some diseases which can cause trouble in families like malignant hyperthermia but you could have other complications from an anesthetic depending on you baseline health anf type of surgery.
Reasonably. Naturally every patient is different and some will have a problem without any previous family history. This is one of the reasons your anesthesiologist is so highly trained to take care of these problems.
Depends. General anesthesia has become safe with all the monitoring that goes along with it. The risk depends on your general health and the type of surgery you will be having. You can talk to the anesthesiologist assigned to your case about specific questions before the surgery as well. Good luck.