Xamarin is a solid choice. It was purchased by Microsoft and is now built directly into Visual Studio. You code in C#. With all the updates and features they are adding, you can do everything but submit to the App Store from Windows, even compile, build and deploy to an iOS device.
For games, Unity 3D is a great option. The editor is free to use for development, and even for distribution (if you have less than 100K USD in annual revenue). Unity supports iOS, Android and most other platforms. It may be possible to use Unity's "Cloud Build" feature to avoid having to use a Mac for deployment, although by default Unity actually spits out an Xcode project when building for iOS.
Other options:
PhoneGap (html/javascript) also works. It isn't quite as nice for gaming, but it's pretty decent for regular GUI applications.
Flutter (dart) is a free cross platform mobile app development framework from Google. Write your code in Dart.
React Native (javascript) is another popular cross-platform framework created by Facebook.
Note that: for all of these options, all or most of the development can be done on Windows, but a MacOS device is still required to build a binary for submission to the App Store. One option is to get a cheap MAC Mini to do your final build.
