How strong must wind gusts be to prevent landings and takeoffs?
You have two questions here, 


  I hope this is not as trivial as it sounds. Is there a wind gusts
  threshold based on which, at a given airport, incoming flights are
  diverted and outgoing ones are delayed?


As far as I know there is no specific threshold for which they will divert. You may want to read up on wind shear here as an airport may advise sever or unsafe wind shear and urge pilots to divert. There will be a point (often at single runway airports) where a landing in a particular type of aircraft becomes unsafe or even impossible as a result of wind but this will vary by plane.   


  Also, I assume that aircraft are designed according to a specific max
  wind gust, so if there is any wind gusts threshold, I expect it to be
  less than the design threshold. Is this correct?


Yes, aircraft are designed with whats called a maximum demonstrated crosswind component. The key here is that this is a demonstrated limit not necessarily an airframe limitation. The number is published in the POH for the aircraft made after 1975 and provides a general guideline of the maximum cross wind the plane can be landed safely in however its more than possible to land it in higher wind situations. 

Also it should be noted that some of this is up to the pilot in command. Just because a towered airport clears you to land does not mean it is safe for you to do so likewise just because a pilot diverts does not mean they were instructed to do so by the tower they may self elect based on their personal abilities.