Would installing a battery 124R with 670 cold cranking amps instead of a battery 121R with 600 cold cranking amps cause problems with my electrical system like my airbag module?

 

It's hard to see how. The battery voltage is the same. But not all electronics likes it when turned off and on. Have you an error indication?

 

My friend and I boosted my car. I took my car to the dealership to get a new battery. My radio worked while driving there and I had no lights on in my dash; I even stopped to get a drink - still no issues. They said they didn't have the proper battery with screw positive posts, so they changed my positive cable and tried to install the 124R with CCA 670. Then they put a 121R with CCA of 600. When I got my car, it wouldn't start. The mechanic didn't have the negative tight enough. When I did start my car, the radio didn't work and my airbag light was on. They claimed this happened due to low voltage. A fuse wouldn't blow for the radio due to low voltage, right? Fuses generally blow for overloads. I'm just trying to figure out how on earth they're trying to say it is my fault and due to low voltage. Any suggestions?

 

If it's only the radio, you possibly need to enter it's security code? This is common with many types to deter theft as only the owner (using the code supplied with the vehicle) can re-enable the radio after it has been disconnected from power.

Otherwise, is there any possibility they accidentally connected the new battery backwards before finally getting in in place right? That would certainly cause blown fuses and possibly major problems with vehicle electronics..

No size of 12V car battery, fitted correctly, should cause any problem with the vehicle systems - though some items may need resetting after losing power.