With regards to the memorial service, check with another neighbour or look in your local newspaper for a death notice. A notice would either state the hour and location of the service, which would mean that it would be appropriate for you to attend if you wished, or it would be listed as "Funeral Private."

Even though you're new to the neighbourhood, your neighbour would likely appreciate your expression of sympathy. You could certainly send her a personal note and even offer to help. In many locales, those who don't know each other well rally to assist bereaved neighbours in some way. Perhaps it's with a gift of food: a casserole, a fruit basket, or some biscuits for the family and their visitors. Sometimes a neighbour offers to house relatives or friends. The circumstances vary greatly, and you'll need to go with your intuition as to whether to do something and what it is you feel comfortable with.