So I tried using a laptop to install Raspberry Pi, without connecting the USB cable to a keyboard and mouse, but I am not seeing any Raspberry Pi display on my laptop screen? Do I need a keyboard and mouse Raspberry pi to work or do can I just use my laptop which already comes with a keyboard and mouse?
You need to set the Pi up to work in headless mode.
First thing is to enable command line access by following the instructions part way down here (section 3).
Once that is up and running, you can enable VNC to allow graphical mode by following the instructions here
All the programs mentioned are free to use with the Pi.
I am not seeing any Raspberry Pi display on my laptop screen
Your laptop is not a video screen.
You cannot display the Pi output on your laptop by connecting them with an HDMI cable for the same reason you cannot display anything on your laptop via the HDMI port.
If you really believe this is an Rpi issue, try plugging a set-top box or smart TV device (or anything intended to be plugged into an HDMI display) into it. It will not work.
I do not know whether it is possible to cause permanent damage by using the connectors wrongly this way, but it might be, so be sure to consider this before applying my last suggestion (or plugging your pi into your laptop again).