There must be a short between two of the down stream load wires someplace. This short could be a short with some resistance to it rather than a dead short.
Things you can try:
Disconnect the wires that feed the down stream outlets at those outlets and the wires to the lamp circuit. This should include unplugging anything that may be attached to those outlets. If this allows the GFI to reset then you know that the fault is in one of those down stream wires or circuits. You could then proceed to reconnect one load at a time in attempt to locate the wire or outlet that causes the GFI to freeze in the fault position.
If, in the initial test above, the GFI still persists in staying reset then you know that there is a fault someplace between the load wire connection of the GFI and the end of that wire (which should not be connected to any loads if you had properly disconnected every load).
The short on the circuits could be a bad outlet, bad lamp connection, bad lamp fixture or bad device plugged into one of the downstream outlets. It is also possible that the short could be caused by a nail or screw someplace that has penetrated one of the wiring cables or the wire insulation someplace has been cut or punctured by the sharp edge of a metal electrical box.
Yes...it is possible that the short that is causing this problem may have existed in the prior situation before you brought the new GFI into the picture.
There is also the remote possibility that the new GFI outlet itself is bad although this seems unlikely due to the tests that you have already performed.