Yes, it is perfectly OK to use your existing MVC pages with ASP.NET Core. In this case, the ASP.NET Core renderer will act as a proxy and serve the MVC pages to the client, so you can have MVC and ASP.NET Core pages within the same project trouble-free. This approach is especially helpful when trying to migrate existing projects to ASP.NET Core.
As for mixing different types of widgets, you cannot place your existing MVC or WebForms widgets on an ASP.NET Core page or template. Only ASP.NET Core widgets can exist on ASP.NET Core pages. Keep in mind that the paradigms of MVC and ViewComponents are similar enough for some code to be reused.