Hi Suzy, I'm not sure I can help, but I can offer some information that may get you started.
The 2007 Fords offered several dealer-installed remote start systems made by a vendor. There were no "factory" systems. There were also many aftermarket companies who offered remote start systems, so you have to determine if you have a Ford system or some other. The Ford systems had a "Ford" label on the remotes, but that label may have peeled off. You can look online to identify the genuine Ford remotes.
First, if the remote light was working, I'd try reprogramming the remote. This matches them to the system.
They system is contained in a plastic box that is tie-wrapped to the wiring under the dash. Ford instructs installers to place it under the steering column area. Among the many wires coming out of the box is a short wire with a black push button on it. This is the programming switch (Ford calls it the "remote start overide button"). There is also a wire with a tiny red LED light. These wires are probably bundled with the others, tied together under the dash. You'll need to contort yourself with a flashlight in hand and look under there to try to find them. Sometimes the programming button is installed somewhere on the dash, usually inconspicuously and low.
That programming button, in conjunction with the brake pedal programs the system; one such programming task is matching (programming) the remote to talk to the car.
If I were sure the battery was good and installed properly (the light on the remote is working), I would try to reprogram the remote. First, of course I'd try the second remote, but you probably only have one.
When you find that button, you can follow these YouTube instructions:
Two types of systems were offered, a "one way" and a "two way". The one way had a button on the rectangular remote that you held and the car started. The two way had an remote with an antenna bump on it. It reported back to you that the car had started.
Here are the instructions for programming the Ford systems. Read the first several steps.
https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=11&ved=2ahUKEwit7JDR7ZLgAhWLdd8KHVjMBxYQFjAKegQICRAC&url=http://machenry.com/documents/Installation%20Instructions/Ford%20Programming%20Instructions.pdf&usg=AOvVaw328EfiUGNhKwPMjAxe1mW2
I know this is far from a "step by step" suggestion to you, but it is all the information I have as I never had to mess with the remote start in my 2008, and without doing it myself, there's not much more that I can offer.
Since these weren't installed at the factory and installed by who-knows-who, it's also possible that the wiring connections weren't made securely and are deteriorating. That would require reinstalling the thing.
The remote can also be bad, so if reprogramming it didn't work, I'd reach out to ebay for another and try programming that one. You can also check YouTube for programming information; maybe someone has a video. The same systems were used from at least 2007 through 2014.
Good luck with this adventure. I wish it was an easier task. Computer nonsense.