While riding the gears shift by themselves. Tried adjusting cable tension numerous times. Does the derailleur look like it is on alignment?

Is the cable slipping?
​​​​​​Shift up thru the gears without pedalling to put tension on the cable. Does it slip? Dont go crazy
Photo looks like a clean cared for bike so i suspect operator error.

No Slipping!

If on rough terrain possible weak deraileur spring or chain to long?

I think that weak derailleur spring is a more likely since I cannot shift into the highest gear. Can I remedy the weak spring?

Did you just put a new chain on there?

No

Talking to yourself? <grin>

Assuming you meant to quote me instead on this question:
It seems from the picture two things I'd want to check about the chain.

1. It looks too wide for that number of speeds. However might be the camera angle,lighting and/or etc,
2. Also looks too long. If there isn't enough tension on the chain from the DR cage then ghost shifting might happen.

What type shifters? Downtube? Might just need to adjust the tension on them. Especially if old friction shifters, but some indexed might move too.

Old XTR 8 speed mountain bike shifters

If its a weak deraileur spring its best to replace the deraileur. A cheap one will do if your on a tight budget. I replaced a nice looking Shimano 600 rear derialeur with a cheap SunRace years ago thinking that I would wait a little and get another 600. Wow... The cheap on is still on there going strong.

As far as the shifters are concerned, I just went through a real tiff trying to rebuild my neighbors old indexed shifters. His problem was intermittent skipping of gears when shifting to a higher gear. I found the problem to be one tiny tooth knocked of of a hard plastic gear/cam. It was not fixable.

My experience is with old road bikes. When I see the abuse gears, chains, and indexed shifters go through on mountain bike, I wonder how they last more than a few months if any...

Title answers your query: you've got a shifty ghost on your hands!

Can't tell if the RD is out of alignment at that angle. You have to look at it directly from the rear. If it's bent even a tiny hair it can cause the ghost shift. As beat up as that bike looks it's possible that it's bent inwards a little.

Those are/were really solid back in the day, they could be sticky. Have you tried flushing the shifters a solvent? I would flush the derailleur also.

I just placed my rear wheel on my truing stand and spun the wheel. All of the cassettes except the last two large ones seem to be wobbling, though there is very little play in them. Can this be a factor in the ghost shifting?

Any wobbling of the cogs will look worse on the larger cogs.

Shimano cogs (and those that mimic Shimano) these days have a series of a few teeth in a row that are differently shaped for shifting ease. These spots can also look to be a wobbled cog.

Many ghost shifting issues are cable friction caused. But weak return springs are also a common issue as the years add up. One way to test for cable friction is to disconnect the rear shift cable (pulling a casing end out of a frame stop is the easiest way)and then move the der with your fingers to see how it works independent of the cable or lever.

There are other reasons like bent teeth, twisted/damaged chain, miss matched chain/teeth wear, badly worn guide pulley. With the wider tolerances (compared to 9, 10, 11, 12...) of an 8 cog system the der hanger alignment is of somewhat less concern, the fewer the cogs the more tolerant the system is to misalignments in general. Andy

The inability to shift to the highest gear (smallest sprocket), assuming that the high limit screw is properly set, also raises the possibility of a cable problem. If the cable is snagging either in the housing or at some other "choke point", like a ferrule, the otherwise-perfectly good RD spring may not be strong enough to overcome cable friction, resulting in sluggish/no shifting to the smaller sprockets. I have seen people try to overcome this by reducing cable tension, resulting in screwed-up shifting across the entire cassette. If your cable hasn't seen some love recently, or even if it has, consider pulling it to ensure that it's properly lubed and not rusting or frayed.

I removed the cable and the only way that I can get it to shift to the smallest cog is to push the derailleur. The limit screw has been loosened.

perhaps a pic of small-ring / small-cog might help eliminate the chain being too long as part (the whole?) problem.

That should resolve the problem.