there is no free version, there is a demo of some sort that afik, can be used like that for months without any problem other than not being able to change wallpaper and colors of the windows and the taskbar, so it will work
then perhaps this could be a option
https://pcpartpicker.com/list/R4H6LD
it went over your limit, doesn't have keyboard and mouse but the monitor has freesync, ideal for gaming with the rx480 i listed there
keyboard and mouse, you can go cheap with a simple combo from logitech or buy a special gaming keyboard and mouse form corsair, logitech or any other brand, there, other users can recommend you better, i personally use on gta v my wireless mouse and my xbox 360 wired controoller, a better combo to play from my bed, other games will ask you to have a better keyboard and mouse and no game controller