The right to repossess the vehicle is governed mostly by the terms of your contract. There is no 3 month non-payment rule. Some lenders may, as a matter of policy, wait that long. But, most contracts provide that as soon as you miss any payment, even by a day, the lender can exercise whatever default rights the contract provides, including repossession.
I'm sure you already know this, but title loans are a terrible idea, in general - way overpriced for the value you get. Sometimes you have to take on in an emergency, perhaps, but this is what happens - you put your vehicle at risk for immediate repossession as soon as you miss the payment.
I am licensed only in Texas. Offering information of a general nature in response to a question is not intended to be legal advice in your state.
No, there is no such rule in Texas. I am unaware of any such rule in any state. Your agreement with the company will govern your rights with respect to repossession. My experience is that these companies will repossess a vehicle as soon as they have the right to do that. You need to read your contract to determine what your rights are. It is possible that there are problems with their contract and/or that you have arguments legally but that would entail hiring an attorney to assist you to determine something like that. It would require the attorney to review your contract etc. My guess is that if you could do that then you would not be one month behind in a car payment. Continue to try to work with them. Pay what you can but do not promise what you cannot deliver. If you need a week to pay then don't promise 2 days no matter how much they press. Your best bet is to be credible and believable about the payments you ARE going to make and stick with that. Good luck.
Remember that most legal issues may be more complex and require more information to answer than what you can provide in this forum. This is not a substitute for contacting an attorney of your choice to get legal advice.