Thunderbolt, Mini Display Port and HDMI I got MDP-HDMI connector that does not work with iMac Mid 2011 with thunderbolt. I do not understand if it should (not) work or it is damaged.

Here is answer 
 that link article  that tells that iMac with thunderbolt should work with Mini Display Port devices.

Here is another answer  that tells that thunderbolt port requires thunderbolt cable they do not say it will not work with mini display port device. I understand that cable is not device. 

It does not make situation more clear. If my MDP-HDMI convertor is damaged or not and I should by another version of connector or my iMac has some problem or I should turn something on to make connector working.

MDP-HDMI convertor has not separate cable. It just plugs in.
Thunderbolt ports allow connectivity to Mini DisplayPort. Since your converter just plugs in and then an HDMI cable (of any length that you supply) is then plugged into your converter and into a display, it should work just fine provided there is no damage to either the HDMI cable or the converter itself. If the Mac isn't recognizing the display, check the HDMI cable or the display first. Try to test the converter on another Mac with Thunderbolt before you deem it dead but Thunderbolt does indeed work with Mini DisplayPort.