I tried to open a communication site in the SharePoint Designer but it shows me the following messages. I have no problem with teams site. So, is there anything that I have to do to be able to open a communication site in the designer? or Designer can not open the communication site at all?
I believe your community site do not have SharePoint designer enabled. Please refer to this old thread to check:
I ran into the same issue. After following Mark L's answer, also open Internet Explorer before you try to connect. I can never connect if i haven't opened IE and logged into your tenant (any sharepoint website).