I'm Doug, and I'm very sorry to hear of your situation. My goal is to provide you with excellent service today.
Maintaining an air filter in a heating and A/C system will not prevent mold from growing in the duct work. Quite simply, mold spores pass through all but the most expensive filters and the culprit is not the mold spores themselves but moisture in the duct work---which is typically causer by excess moisture in the air or a cooling system that is too big for the space being cooled.
Likewise, a simple carrot will not prevent a normally operating disposal from functioning, and the suggestion that you are responsible for the breakdown of the disposal based on a carrot being in it is patently ludicrous and a falsity.
As for the mold problem, contact the local county health department and file a formal complaint. They should come out and check the mold levels and see if there is any toxic mold present that would dictate that the landlord must remediate the problem---at the landlord's expense. You should not have been charged for the disposal repair and you may sue to recover the expense if they will not voluntarily return the fee they charged for routine maintenance---again, which is their legal obligation to provide at no cost to you.
You may reply back to me using the Reply link and I will be happy to continue to assist you until I am able to address your concerns, to your satisfaction.
Please be so kind as to rate my service to you. That is the only way I am credited for assisting you.
I wish you and yours the best in 2016,