Is there any option to avoid the appearance of profile icon?

Yes. In the desktop app. you can hover at the bottom of the circle and click the X. This isn't a just profile icon though, it's also the camera view. You can chose to record the camera and screen or just the screen. Picking just the screen will make that circle go away as well.

Yes. When using the chrome extension there should be a small X in the upper left corner outside the profile image circle.