Having diabetes, age 73, should my nurse continue to give me insulin injections when my sugars are below 60?

Maybe not then. Your questions appears to lead to a logical response of no. But if you are felling well, and you normally need Insulin everyday, and you are about to consume a large meal, the answer would probably be yes. You should get some insulin, as you don't want your blood sugar level to shoot up too high either.
No ! Hypoglycemia at this age can be very dangerous and can induce arrhythmia, hypotension and decreased cerebral blood flow. As much as good diabetes control is needed, it should not be too intensive. If your bs tends to be low frequently then you need adjustment in your Insulin right away. Hypoglycemia (below 70) should always be corrected first before any decision is made to give insulin.