the dust that is seen in your images on the side seems to get under the digitizer and the lc screen. I've taken my L920 apart many times to change the housing between black, yellow and red (
http://forums.windowscentral.com/nokia-lumia-920/225333-7.htm#post2169946 - check out the video by LE55ONS on how to take it apart for repair). Never did anything like this happen. It has nothing to do with the tightness of the housing but the rubber seal that sits on the side of the screen itself (if you take the housing off you will see the rubber seal stays around the screen).
If I had this happen to mine, including the crack at the top I would try to replace the digitizer glass or better the entire screen.
nokia lumia 920 screen replacement | eBay
it does seem straight forward to replace the entire screen (lc plus digitizer) and they are somewhere between $70-90.
Nokia Lumia 920 Front LCD Display Touch Digitizer Screen Assembly Repair Parts | eBay
great screen repair guide by LE55ONS
Nokia Lumia 920 Digitizer Glass & LCD (TFT) Display Replacement - YouTube
he is taking the screen apart here and this is not an easy job because the digitizer is glued to the lc with adhesive that can be very challenging to remove.
other
Nokia Lumia 920 Screen Take Apart Repair Guide - YouTube
Altogether I'd decide between the price of the company you mentioned and replacing the entire screen. That is totally up to you. I would not see a problem replacing the entire screen myself, however, I'd stay away from replacing the digitizer alone. If the company is e.g. $100 it's probably a good idea to have them do it. if more expensive you need to evaluate.
But, if you do decide to replace I'd first try to take the housing off and then carefully remove the seal from the affected side and try to suck the dust out with a vacuum cleaner that you extended to a small tube or hose that fits over the area. Then re-apply the seal. That of course will not take care of the crack.