Yes, the speed, and yes the lights: green for 1G and amber for 100M.
The cable was the first thing I suspected, so I did try short lengths of cat6, over the carpet of course, because there's no flat cat6, I think. Same lost pings. But the flat cable worked just fine between two units if I bypassed the router with a cable joiner (and set fixed IP addys). Lost pings also between two Raspberry Pis via the router at 100M.
1000BASE-T was originally specced to run over 100m of the original cat5 cable (not cat5e or cat6), suitable for existing installations in large businesses, but it did push cat5's safety margin somewhat and it would become unreliable as the plugs oxidised and the cables aged. So cat5e and then cat6 tightened the spec to increase the reliability.But there should be no problem for newish cat5e over the relatively short runs of a home installation for 1G. The flat cable that I bought (forget where) is actually marked "Cat.5E Verified for gigabit Ethernet".