Try connecting the cable box to HDMI 2 which you know works with the Firestick.
If you get a picture than HDMI 1 and 3 are probably defective.
If you don't try a different HDMI cable. Even if the cable worked with your old TV doesn't mean it will work with the new one.
Because you don't get a picture with both the cable box and PS3 I will guess that the TV is the problem.