Answer to "Why does my Atari show a blank screen when turned on?"It could be the RF module. Another solution could also be necessary to make adjustment of the contacts which hosts the game cartridge. Clean and adjust.
Answer to "How to fix the left bumper"Look at the front of the control cabinet. In the part where it rests on bumper, there may be attrition. Also the micro-switch may be damaged
Answer to "How to repair a loose analogstick?"You need to change the potentiometer. You can get a potentiometer in good condition, by an original xbox control.