No. They do not. Current guidelines suggest that a reaction in a family member not be considered a problem w/ vaccinating, and no study has found a link for serious reactions among family members.
No. Although anyone can have a reaction to a vaccine, there does not seem to be a higher chance of serious reactions in relatives of those who had serious reactions themselves. Common vaccine reactions: pain (for almost all), fever (in some), swelling of the vaccination area (in a few) and swelling of the entire limb injected (very few). All these reactions vary depending on age and type of vaccine.
Yes. ....But not the way you might think. Allergies to foods (and thusly, vaccines) are most directly reported between siblings, not between parents. Any concern regarding an allergy to a vaccine should be discussed with your provider. For the purposes of vaccines, a hypersensitivity should be considered to be equivalent to an allergy.
No. There is no evidence that these run in families.
No. Serious vaccine reactions are almost unheard of and certainly don't run in families.
No. In my years of practice, I have not seen vaccine reactions run in families. Vaccine reactions appear to be uncommon and random. If a particular family had many members with hereditary weak immune systems, then those members might get reactions to live vaccines such as the chicken pox or measles vaccines.