Call me crazy, but this is not your problem. As the tenant, this is well beyond your scope of responsibility (though you should try to keep it propped up until it’s fixed since the dogs are your responsibility).
Write a letter (certified mail) to the management company and the owner, ASAP. Tell them about the dogs and small children, and let them know that you’ll be happy to deal with it, but will be deducting the cost from your next rent check. Give them a date by which they need to respond before you take corrective action, maybe 15 days from the date you send the letter.
Be sure to include contact information for yourself and let them know what transpired with the management company (they should indeed be responsible for these repairs, IMHO, or at least they should be responsible for communicating with the LL if you were never given the LL’s contact information). Tell them that you need the LL’s contact information since the management company is not cooperating on repairs. Send the same letter to both the mgmt. company and the owner and “cc” it so they both know.
Hire a licensed fence contractor so that the owner cannot fault you if the fence isn’t fixed correctly or if it comes down again, prematurely. Let them know that you will be hiring a licensed contractor if they don’t take action to fix the fence immediately because of the liability issue.
And yes, the cost of fence repair/installation should be shared between both owners unless the fence is entirely on one side of the property line. In that case, it’s the owner of that property’s responsibility, but it is also 100% their fence. You cannot attach anytlhing to it, cause damage to it, etc. I would not recommend going this route. Try to do a nice “good neighbor” fence directly on the property line and have the **owners** share the cost with one another.
