Is it possible to play Amiga games on a PC with Windows OS, and is it also possible to connect an Amiga type joystick to PC and play the games with it?
Get WinUAE, an Amiga emulator for windows.
The Amiga Joystick can be adapted to the PC paralell (printer) port with some fancy wiring, but getting it to be a "real" windows joystick is going to be a challenge. You would probably have to write a device driver to replace the default parallel port driver.
I have heard rumors that Atari Age used to offer an adapter that would work, but it looks like it is discontinued.
Yes, it is possible to play Amiga games on a Windows using the WinUAE emulator, the official site is not the one mentioned in the accepted answer but is actually http://winuae.net.
It is important to note that if you do not own an actual Amiga, using the system ROMs without license is illegal.
The legal and practical way to emulate an Amiga is to use the Amiga Forever package by Cloanto which has an official license to bundle the official ROMs for every single Amiga released. It comes with the aforementioned WinUAE emulator and a number of free software, games and demos.
This is by far the simplest way to legally run Amiga software on a Windows PC.
Note that after Amiga Forever is installed, you can freely use use your own downloaded version of WinUAE or FS UAE (an UAE derived emulator which is multi platform) if you do not like the default interface or wish to test a newer version.
Yes, it is possible to use an Amiga to USB converter (you can find one on the site AmigaKit, just google it).
Note that you are better off using a regular PC joystick or controller since Amiga joysticks are generally not very ergonomic. Both WinUAE and FS UAE support all kind of PC, consoles joysticks and controllers so you should have no problem using a regular controller with them.