In order to do the following, you should have an icon of Firefox on your desktop. Place your mouse pointer on the icon and click the right button. Choose properties from the pop-up menu that opens. Click on the Shortcut tab (if not already displayed) on the top area of the pop-up window. At the right side of the Run option, choose Maximized instead of Normal window. At the end press the OK button at the bottom of the window. If you want to place the Firefox icon on your taskbar, than click and hold the left mouse button on the icon and drag it onto the taskbar.
I think the fact that the pointer moves from the bottom of the screen to the top of it when you try to move the pointer below the bottom edge is due to a mouse setting. My mouse...