Maximum current vs temperature
The exact temperature rise depends on the thermal resistance from the wire to ambient, which we don't know.  This depends on insulation type, thickness, and whether the air is moving or not.

However, looking at the dissipation per meter or per foot will give you a conceptual feel.  If that's low enough, you don't have to do the formal calculation.  You say your first cable has a resistance of 161 &micro;&Omega;/m (I'm using your figures without checking them).  At 175 A, that is a dissipation of 4.9 watts/meter, or 1.5 watts/foot, or 125 mW/inch.  

Imagine a one inch length of wire in front of you.  125 mW sounds like it would get barely warm for such a decent size chunk of copper.  To put this in persective, you can have a SOT-23 package on a PC board dissipate about that much safely.  The one inch chunk of cable will be much bigger.  At this low level of dissipation per length, you are going to be quite safe and further detailed calculations aren't worth the effort.

However, if you really wanted to calculate the temperature rise, you'd somehow have to come up with the thermal resistance of the cable to ambient.  That would be specified in units like &deg;C per watt per meter of cable.  Multiply that by your 4.9 W/m value, and you get the temperature rise of the cable above ambient.