Why is my ABS light on?

The light is not going to go off until you repair the issue that triggered it in the first place. The first thing you need is the code number being set in the ABS module. It sounds like you may have a wheel speed sensor that is out. These videos will help us find the problem.

https://youtu.be/rTtAnsOlZU4

and

https://youtu.be/Pb6nXV-iamI

and

https://youtu.be/ANWVTh2RQoQ

and

https://www.2carpros.com/articles/abs-warning-light-on-easy-repair-guide

Please run down these guides and report back.

Since the light is on, have the computer scanned for trouble codes. That way you will know where the problem is coming from. You can also try disconnecting the battery for about 20 minutes to see if that shuts if off. However, I would start by scanning the computer to see where the cause is coming from, then I would check to see if something is disconnected, damaged, dirty.

The ABS light really has nothing to do with the pads and rotors. It means the ABS computer has sensed an electronic malfunction, shut the system down and stored a trouble code to identify what it saw. You need to have the ABS module scanned to retrieve the stored code and that will give you a starting point. My guess is that you will find a problem with one of the wheel speed sensors, possibly damaged during the brake repair or maybe a brake rotor with a missing or defective sensor ring.

What did you replace? Pads and rotors? The normal reason for those lights are a bad ABS sensor or bad sensor wiring. What you need is a scan tool that can read the ABS codes and the actual speed data from the sensors. That way you have a place to start from and a way to confirm that it is repaired.

The light being on just means no ABS. Yes either a good shop or dealer. The parts stores usually do not do ABS scans because of the liability concerns.
As long as the brakes work okay it is not an "oh my god the car's going to blow up" thing but I would get it looked at when you can, not having ABS when you are used to it can be an awakening moment.

Hello and welcome to 2CarPros.
That sounds like you have a problem with the cluster and not with the rest of the vehicle. When you first start it and it runs the bulb test do the lights come on bright and go off as they should? Unfortunately you need to have a good scan tool that can actually "talk" to the cluster to test it and see if it's internal to the cluster or if there is a communication error on the network.

When the lights fade in/out does anything else happen with any other part of the car?

Okay, so they fade on/off in a regular cycle? Does this happen when you just turn on the key or does the engine need to be running? As a test you can go to the under-hood fuse box and look for the ABS module fuse and remove it. If the problem is in the module or related to it the ABS light should change. If it does but the CEL still acts like a disco then it is more than likely a cluster error. Ford uses the cluster to do a lot more than just the gauges so anything can effect them. The attached diagram shows the possible fuse locations depending on the build date. Ford made a few changes depending on model and engine. The cover should tell you which it actually is.
For the brake pulse, I would pull the wheels and then the rotors, clean off the inside of the rotor and the surface of the hub, then carefully tighten the lug nuts to torque and see if it stops, a tiny bit of rust or dirt on that mating surface will cause the rotor to have serious run out at the friction surface.

That indicates that the corresponding control modules have recognized a malfunction in the system and set a trouble code to identify the issue. You will need a professional scan tool, not just a generic code reader to retrieve the codes and that will give you a starting point.

That depends what the problem turns out to be. The diagnostics average about $100