Output Characteristics : Common Base vs Common Emitter
Your top graph seems wrong. Either the horizontal axis is mislabeled (it should be Vce for Common Emitter) or this is not a Common Emitter curve trace. Also the knee voltage is very high for a bipolar transistor, and the Collector-Base leakage current is appalling.

Here's a better example of a Common Emitter curve trace:-


The reason Collector-Emitter current goes to zero in common Emitter configuration is that the Emitter is connected to the lowest voltage in the circuit. As Collector voltage approaches this common point junction resistances start to limit current, eventually reaching zero when it cannot pull down any further.   

In a common Base circuit the Base is connected to 'ground' but the Emitter is powered from a lower voltage. Therefore the Collector is able to go below zero volts as it is pulled towards the Emitter. 

In both cases the transistor will saturate when Collector voltage approaches Emitter voltage and current is limited by resistance. As far as the transistor is concerned it is still acting the same, only the reference point has changed (from Emitter to Base).