Well, we already have the PraiseDB:
It is more a piece of software that helps users add mods to their games than a website for communication and tutorials, but it handles the browsing, downloading, and uploading features you suggested.
Yes.
I feel it is perfectly OK to make an announcement about the site here, and post major update news as you feel appropriate. Mentioning it in some relevant posts is likely also OK. Posting links to your website all over the Discourse in every 3 topics is not OK. I trust that you can find/understand the difference between encouraging people to visit and spamming. (and the moderators will certainly let you know if you don’t 
).
All that being said, I am curious what you feel a dedicated website will do that the Discourse cannot. While I do not pretend that Discourse can do everything, I am having a hard time picturing what a dedicated site would do for people, when a mod upload/download site exists already, and lots of modding discussion/tutorials happen here. Don’t want to get to a place where a mod creator has to know the X sites to post there mod on!