The green light just shows that the modem sees SOME signal on the inbound side, not that the signal is valid or that the speeds are fast. Trying to use home wiring for networking is dodgy at best. Any static on the power line will interfere with the signal trying to get through. Remember, those interfaces are designed to use the line ONLY when the voltage on the line from the alternating current is close to zero, which occurs either 100 or 120 times a second, depending on the frequency of your power system. But that time interval is VERY short, because as the voltage rises on the other side of that null, the interface stops trying to communicate. If there is a bit of static on the line, or if there are competing signals from other devices trying to use that tiny time slice, even in a neighbors house, your signal ma be interfered with and you see that as slowdowns and eventually disconnects.
Is there any way to connect to the ISP modem without the Homeplugs? Have you tested that to eliminate the Homeplugs as the culprit(s)?