If there exists a cascade of op amps, does the previous op amp's slew rate affect the later op amp's slew rates?
Do the slew rates combine in some way? 


YES


  Is unity gain instability the reason for the second decrease in slew rate?


NO


  Is there a variable that affects slew rate I am not considering?


YES
Follow the recommended Figure 1 test conditions in the datasheet then understand why.

Slew rate is current limited into either Miller capacitance or load capacitance or limited by excessive low load resistance or any combination that reduces drive current available to create the virtual ground at the input.

Normally this ancient 1MHz uA741 style Op Amp, the LM348 is expected to have a nominal rise time of ;

$$t_R=0.35/f-3dB=0.35us$$
This is defined by the 10% to 90% slope of a small step pulse for a "small signal" response.

Max Slew rate spec of 0.5V type will reduce when cascaded by equal devices. This is a 'large signal" response and is limited by the current limited output and load.  The datasheet defines it for 2k//100pF to achieve 0.5V/us large swing.


If you choose 50 Ohms for R1 and R2 then the output will be driving R2 as the load for +/2.5V then it needs 2.5V/50R=50mA.  which is current limiting the OA.
If you used 500 Ohms and 5V then it would demand 10mA and the balance of available current of 15mA into 100pF can drive is as follows;


$$dV/dt = Ic/C = 15mA/100pF = 0.15V/us$$

This is key to understand how current limit reduces slew rate from R as well as C load effects.  

The proper way to design is to define the input, output characteristics and impedances then choose the best parts to meet that criteria, rather than choose whatever is around and guess on the wrong R values to use.

A 50 Ohm driver with 5V square wave needs at least 10x the BW as the applied frequency to have the 9th harmonic and be able to source the current to drive the load Vp/R+ CdV/dt (load capacitance).  

So I suspect the LM348 is not a good choice.  

In future, specify output slew rate required, Voltage and load impedance. Then consider BW, drive current, load resistance and cable capacitance in your design. ( e.g. 100pF/m )