Sandy - you always seem to have the most awesome questions! Sorry to hear why though. As I understand it, you have the following:
1. Deferred Maintenance caused by leaking HVAC lines;
2. Roach infestation;
3. Unhappy Residents (rightfully so)who have turned to becoming vigilantes;
4. Unwanted violations from the City resulting in a PR nightmare.
The good news is everything is fixable! The opportunity is there to come out on top - If Ownership is ready to get it done. Are they?
I think the first thing is to evaluate each unit for damage and get a game plan in place. You may or may not know this, but when you are talking MOLD, it cannot be just handled by onsite maintenance staff unless they are specially trained and certified to remove contain and remove it. It takes special gear. You will have to hire an outside firm. Once this is done and arrangements made, I would go directly to the City myself and ask for a meeting and have the contract in hand for the work to be completed (start date and approximate completion date.) You'll have to attend the hearings, of course, but I would want your company attorney to attend as well.
Second, while working on that, I would open the office. Where are these residents supposed to go to seek help? It sounds like they may have less income to buy gas or take a bus to go across town to talk to Management. Even with some restricted time/days open, at least the Residents have easier access. You should notify all residents in as many ways as possible what you plan to do to handle routine and emergency work orders, including online access. Not only is this necessary, it will help prevent an issues with Fair Housing should someone's work order remain undone or put off too long.
Next, I would invite this group to meet in the Office and I would ask to address the group and start a conversation. (Not that this will be pleasant, I know.) If they decline, at least the Invitation was made. Once you start addressing the maintenance issues this group will probably go disperse.
Roaches are attracted as all pests are to a source of food. Period. Implement immediate housekeeping inspections in addition to getting the entire building treated. Roaches can be difficult to eradicate but it can be done. I had an entire property so infested when I got there that the roached hung in clouds clumping in corners of ceilings. Gross does not begin to paint the picture. It was resolved through this kind of program in much less time than you'd think. Stubborn problem? Yes, but within 4-6 months, the problem was resolved. We threatened eviction for non compliance and brought in housekeeping experts to teach the miracles of bleach, vinegar, baking soda and ammonia with elbow grease.
Then there is the PR. The only way to make it better is to make your efforts known to others. You might have to toot your own horn there and start with holding some resident activities and invite others from the community to join in. Having a neighborhood watch group might be a start - then maybe your Ringleader who complains can spearhead this group instead and put that energy to a more PR friendly use.
Once things start getting better, hire another person - Assistant Manager who can and likes to lease. You have your hands full. I hope your Owner is on board. Finally, if there is not a vacant to put that one family in, then the Owner might need to step it up and in my opinion, he should relocate the family to somewhere else at his company's expense. Health problems is a liability and I feel I would want my residents to know their health matters to me.