Rear derailleur shifting issues -- replaced shift cable

Shift cable is corroding.
None of the cable inside the cable guide was separating but the corrosion was causing the cable to swell and not move smoothly inside the housing (was getting stuck).

The cable hold-down bolt has a tabbed washer that's incorrectly positioned also (although this didn't cause any issues).

The short housing between the derailleur and the chain-stay tube is in OK conditions so just cleaned it and reused it.

After installing the new cable (with the shifter in the smallest gear position), used the cable adjuster (arrow) to adjust the derailleur for shifting.

These two screws are the high and low limits for the travel of the derailleur.
These are adjusted so that the chain doesn't gravel too little or too far at the end of the derailleur's travel range. 
It it goes too far, the chain can come off the gear.
If it doesn't go far enough, it won't engage the two gears at the end.

The free-wheels are not too worn.
Don't have to replace them yet (I've replaced them before).

Cut the end of the shift cable to length and just before putting the cap on the end of the cable.
Will test out the shifting & the new chain on my next ride.
The derailleur may need more adjusting, or not...

 

Some of the chemicals used to clean and lube some of the components that the chain goes thorugh.

Some of the tool  used to clean and adjust the gears and derailleur.