Start by setting the members of the teams as either staff or volunteers and then assigning tags to the members of the teams. (One tag for each neighborhood, etc.)

Once you have marked the people as being on the teams, you can create lists or filters with the people for the team. (Lists are static and will only ever contain the people you put on them. Filters are dynamic and will show the people with that tag at the time you apply the filter.)

For non-volunteers, you can set up the same kind of list based on the address. You can then map the addresses on any given list by going to the list and clicking the map button for the list. 

For users to communicate with the team, you would set up a broadcaster for the team with an email address. When people send an email to the address, their profiles can be given an automatic follow up for the team to check.

With a bit of Liquid coding, you can set up pages in the site to show events from a certain neighborhood and have people who RSVP to those events set to be followed up by someone from the team.