"Corruption" errors on startup of Task Scheduler

[IainB]

This has been posted in the hope that it might be of help/use to others in fixing what looks like it could be a relatively common problem.

Description of problem:
The PC is an HP ENVY 14 laptop running Win7-64 Home Premium.
For a while now, I have been getting some odd startup error messages on this laptop - nothing serious or fatal - and today I got another one: the HP Support program popped up a message window saying that it had an update task that required a restart. So I restarted.
After restarting, it asked for a restart!
After restarting again, it still asked for a restart.

I figured that the Task Scheduler might have the details on the task, and decided to take a look as the task clearly wasn't working as it should.
I have to admit that I haven't really looked in the Task Scheduler in ages, and when it started up I was a bit alarmed to see that it had 4 errors that popped up one after the other. I closed the Task Scheduler and started it up again, and the same thing happened. I repeated this a third time - same result. So here were a bunch of consistent errors that indicated that the Task Scheduler was probably unable to work properly and that it could have been this way for some time, without my knowing.    

The errors:

[attach]

[IainB]

It turned out that I had been somewhat premature in thinking I had fixed all 4 errors in Task Scheduler, just by fixing one. Following a laptop reboot the three remaining untreated errors returned, so it seemed that I had only fixed the first error.
So I have made a major change to the OP above, covering the detailed process for fixing the remaining 3 errors.

Task Scheduler is now running properly. I think it had been unable to run properly - if at all - due to the errors. This might explain some of the startup oddities I had been experiencing, as the Task Scheduler is invoked at startup (as well as at other times). The problems had apparently also frustrated the proper update of some standard software (HP Support Assistant), which completed its update just fine after I had fixed the errors in Task Scheduler.

[cranioscopical]

It was thoughtful and patient of you to document that. I rely on task scheduler a lot for my own alarms, apart from the system requirements 

[IainB]

@cranioscopical: Thanks. Though I did find some useful information in the references that I gave, I couldn't find anything quite as helpful step-by-step as this, so, whilst I was writing my own notes in OneNote about what I did (whilst it was still fresh in my memory), I thought it could be useful to record it as a bit of knowledge for others - it might be a common type of problem.
I suspect that many people could have similar problems with the Task Scheduler, but might not actually know it, yet.
I mean, it was a surprise to me, and then I wondered how long this had been going on. I could have remained blissfully unaware of any problems had I not started up Task Scheduler in the first place. Like I said:

I haven't really looked in the Task Scheduler in ages...