I would get a lug swaged on the end of the cable and bolt it right on where the bolted junction is made in the left side of the picture. The wire rope clip looks like something a farmer would have done. An ordinary bolt will be fine, but use sandpaper to cleen up the strap where the lug contacts it. If that strap is aluminum use the copper to aluminum electrical joint compound on it. #2 cables 25' long would be fine, it helps to make the electrode cable a bit longer than the ground [20' ground & 30 electrode ?] as that is the one that needs freedom of movement.

For 25 feet:
#4 is actually fine too, but it limits your duty cycle, but the wire -may- still have a higher duty cycle than the machine.
#1 for 180A should give you a duty cycle of around 100%, which is most likely more than your machine (my guess is its 100% at around 130A, right?)
Charts I have seen, show #2 at a 60% duty cycle for up to 100 feet (combined stinger, and ground length) at 200A, and #4 at 60% for 100 feet at 150A (so maybe 40% at the full capacity of your machine).
The advice about the electrode being longer than the ground is good.  Also, lighter electrode cable is less weight on the stinger.
I've actually got #2 on my current welder, but when I had #4 on a previous one, I had no problems.

I also put spiral wrap around all but the last 10 feet or so of my electrodes, to keep the insulation from being damaged as it gets dragged on the floor.  Don't know if anyone else here is that nuts/retentive.     :)

Yes, that wire rope getup is just funny.
Personally, I would get a lug crimped onto a foot or so of cable, and install a quick connect hanging from the machine.
Any bolt should do (although I usually choose a stainless one for the extra few cents).

As for the switch, I would suggest dielectric grease.  Available at any auto-parts store, its used to lubricate and prevent corrosion on electrical connections, such as bulb contacts, switch innards, and spark plug boots.  Really good stuff to have around, and its also good on o-rings.