I believe the speed label in this property window is intended for the port speed of your eth0 network device, and not for the 'speed' of your internet connection.
Your internet 'speed' can be interpreted many ways. You get headline figures from your ISP, but you probably never see these figures because your connection is contended. Here in the UK, a 20:1 contention ratio for home broadband ADSL is typical, so your line is split between 20 people as a worst case scenario at peak times.
Other factors that affect your broadband speed are limited connections upstream, and perhaps even bandwidth shaping, as well as network latency and jitter.
Quite simply, you can't "get Ubuntu to do a speed check."
To answer your other question, Ubuntu will not collect information from your visits to the various internet speed check sites, and put the results beside the speed label.