To get 144 Hz at 1080p, the computer and monitor must both support HDMI 1.3 or higher, and the monitor must support 144 Hz over HDMI (not all monitors support it even if they have higher version HDMI ports). 1080p 144 Hz does not require HDMI 2.0.
The cable must be a High Speed HDMI cable (340 MHz / 10.2 Gbit/s rating) or better. HDMI cables are not classified by the same "version" system that HDMI devices are since cable do not have any effect on version support, only speed support.
Yes, the monitor will be able to work at 144 Hz without any special modifications, just plug it in. At worst you will just need to open Windows settings to change the refresh rate (Google it).
No, having HDMI 2.0 will not interfere with any operations...