You'll need to call your car insurance agent and discuss part of what you ask for. When attached to your car or truck the liability to others is automatically extended to the trailer. But... if you turn and bite a tree/pole with it and damage or destroy the trailer.... not covered. You need to add comprehensive to the trailer itself to cover you damaging your trailer.
Triple A (AAA) could cover you for tire changes, but really..... go buy a second spare wheel and tire, go to harbor freight and get a small bottle jack and carry a Harbor freight telescopic 1/2" ratchet, 4" extension and deep well 6 point sockets to fit your trailer and your truck lugs.
As for break downs.... I don't think you'll get an insurance for break downs unless your eligible for an extended warranty. Just wait for your cell phone ring tomorrow and discuss that concept with the spam caller who calls.
I do keep comprehensive in my 3 trailers through Erie Insurance (my life long car insurance company) and it is comically cheap. My cheapest trailer is around $3K to replace and my dump is around $10K to replace, so justifying spending $100 or so a year to add comprehensive is easy to do.