I would recommend constructing a HTML5 Web Page with the components that you desire, there are many such components available, e.g. Clocklink has a number of really short HTML5 pages that can be used to add an analogue or digital clock, e.g.:
can be added to a web page by pasting in a link 288 characters long from the site, (code not shown as this would be redistributing the code to people who haven't seen the T&Cs).
Similarly, Accuweather and a number of other services offer an HTML code to paste a variety of weather components to your page:
Once you have a page that contains all of the information that you require simply open it in a browser, drag it to the second screen which is your projector and make it full screen.
This will work in any browser on any OS as long as there is an internet connection and the browser supports HTML 5. While there are Flash equivalents they will restrict your choice of browser and OS plus may constitute a security risk.
Given the platform independence you could also consider connecting something like a Raspberry Pi to the projector so as to avoid tying up a PC/Laptop full time.