Aren't most of the Shimano cheap derailleur series spring loaded to jump to the smallest cog?
Yes, an internal spring retracts the derailleur toward the smaller cogs when cable tension is decreased.
I understand the limiting screws but I have never worked on the index shifting. Is it a single cable that ultimately should be the perfect length and not under tension when the chain is on the bottom cog?
Not quite. The cable is always under tension when it shifts from 2nd highest gear through the lowest gear. The only time the cable is not under tension is when it is in the smallest cog (high gear). With the derailleur set to the center of the smallest cog the cable is pulled snuggly by hand tension to take up the slack and then secured in place with the pinch nut on the derailleur. Yes, correct, the cable is not under tension on the smallest cog. Maybe a little tension, just snug usually.
A slow lazy shifter tells me there is slack in the cable and you are pulling up that slack every time you try to shift to a larger back sprocket.
Yes, there should be no slack in the cable, just no tension on the cable only before shifting out of the highest gear, then there is always cable tension as you shift to all the lower ratio gears.
Then when the default return spring of the derailleur tries to pull up the slack in the cable your chain is now slightly misaligned.
Chain should not be misaligned with the cog. Several causes: 1. cable has too much friction inside the cable housing not letting it move freely, causing cable to not return to its intended position. 2. Lack of lubricant in the derailleur parallelogram pivots causing friction and not letting it move freely. 3. If not centered in the highest and lowest gear, the limit screws are not set properly. 4. The B screw is in too far causing the cage to be too far away from the freewheel causing the shift to happen late.
The cable barrel adjustment seems to just take up the slack in the cable.
Yes, plus it is used to center the derailleur cage pulley below the freewheel cog. Usually set when the bike is in the middle gear if the cassette/freewheel.
With Shimano Index Shifting (SIS) the shifter and derailleur have to be SIS specific. The ratios of movement match each other. Also most of the shifters have indexes that click as the derailleur moves to each gear center line.
