Hi and welcome to the forum!
The ADF games are usually meant to be run without AmigaOS desktop running in background. The best method how to run & play ADF games is to run the WinUAE configuration window and specify the path to the ADF files in "Floppy drives" section. Also don't forget to adjust the emulation to the game requirements (A500 vs A1200), etc. For more info please check the chapter named "How to use an ADF file" of this website: http://www.mameworld.info/easyemu/winuaeguide/uae-adf.htm
Downloads folder can be found in AmiKit:Internet/Downloads folder.
WinZip, as the name suggests, is a Windows program and as such it won't work in AmiKit, of course. Check out similar program in AmiKit environemnt, eg. MUIUnArc or VooDoo-X. Click the green eye start button in lower left corner of AmiKit, select Tools and there you can see a section of programs for un/archiving.