Can I connect existing USB mouse and keyboard and VGA monitor to my Nexus 6
A plug-and-play USB keyboard and mouse should work fine with your phone.

The Motorola Nexus 6 does not support video out of its microUSB port with any adapter/protocol. 

You could cast video from the phone to a Google Chromecast ($35 MSRP) HDMI dongle, and then purchase an active HDMI female to VGA male cable/adapter to connect the Chromecast to your monitor - but note that the Chromecast and phone would need to be on the same WiFi network, and there is slight latency in wireless casting that you wouldn't get with a video cable.