Did this aircraft illegally exceed 250kts below 10,000ft?
First of all, this aircraft was flying in Europe, not in FAA land. In Europe there is no general limitation to 250 knots below 10000 ft. There is no EASA airspace restrictions, every country has their own set of rules. There are airspaces that have speed restrictions below 10000 ft, others don't and sometimes aircraft are allowed to go faster than 250 knots with ATC permission.

Secondly, the 250 knots is an indicated airspeed limitation. At 10000 ft, 250 KIAS (Knots Indicated Air Speed) will be approximately 289 KTAS (Knots True Air Speed). And then there is the additional effect of wind. With a strong tailwind you can exceed 350 knots ground speed while still flying below 250 KIAS.

Thirdly, the data in the image you linked is provided by ADS-B. ADS-B can transmit either ground speed or air speed. Under the European ADS-B mandate, aircraft must transmit ground speed, not air speed. But since this mandate will not affect this particular aircraft until December 7, 2017 it might very well be transmitting True Air Speed. 

Note that ADS-B data is not purely derived from GPS. The Mode S transponder which provides the ADS-B out functionality on this aircraft is attached to various data channels and is getting Indicated Airspeed, barometric vertical rate. In fact ADS-B vertical rate is preferably from a barometric source.