QNetworkAccessManager is the Qt component for make network connection. The use of this component is quite easy and you can use for basic operation like download files or web pages. Experimenting the use you'll note it work very well with standard network connection but, especially under Windows, doesn't work with SSL connection (like 'https' for example).
Convert a SVG image to QPixmap format is a very easy operation using the Qt classes. Common examples that is possible to find into documentation, however, refer to a situation where the screen device have standard pixel ratio of 1.
If you want to develop a custom Qt widget to distrbute or sell a good habits is always to develop, in addition, a plugin working into Qt Designer allowing to visually use and configure your custom widget inside the designer GUI editor. Develop such plugin is not a difficult task, official documentation and examples are here and, as usual for Qt, all is well explained.
Another important task of software development is to create a good graphic elements for make the interface look better. Graphic texts, icons, logos and so on are all parts of a good product. If you, like me, are not so good in graphic design we need some help...