You decide what your question is in the first place before using your keyboard. In that post you typed - "n now there are even more sports events to be broadcasted ...namely Wimbledon, World T20 qualifiers,Ashes,Nz tour of India (Women)"
All these events are when the sa vs ban series is happening and not till 2020!
So if your question is if the sa vs ban series is being shown, the answer is yes as per the mails and screen shots already posted.
If your question is does Star have the rights of Bangladesh home cricket till 2020 then the answer is yes.
But if your question is will Star show all the matches happening in Bangladesh till 2020 then no one can confirm that to you at this point of time as it all depends on priorities of Star Sports.
FYI, Star Sports had the rights of all 380 matches of EPL, La Liga and Serie A last season but did they show all the matches live on their channels? The answer is no. So there is a difference between having the rights of any league or cricket board and showing all matches of that league or hosted by that cricket board.
Just because Star Sports didn't show all 380 matches live of EPL, La Liga and Serie A last season did not mean that they didn't have the rights of that league. Similarly, just because Star Sports didn't show the Bangladesh vs Zimbabwe series last November doesn't mean they don't have the rights of Bangladesh cricket.
Hope this explanation in simple English is understood by you. If you are alive till 2020 you can see whether all the matches hosted by BCB are shown on Star Sports or not and that can give the answer to your question!!!!!