(1) CHKDSK examined your clusters/sectors in partition 5 & found no problem. Had it found a problem, it would have attempted to recover any data & put it into a .chk file(s) (written onto partition 5, maybe in a "Found" folder). Also, it would mark the sector(s) as unusable. Then Windows wouldn't use it for folders & files.
I'm thinking CHKDSK, itself, wouldn't consider a sector(s) already marked bad in its next run. So, there may be bad sectors marked unusable from a prior run (Windows might even run it automatically) -- that maybe Acronis does still see. I see in my DiscWizard .pdf there is a way to have Acronis ignore those errors. Maybe just do that.
I also guess the HDD itself has firmware aboard that marks sectors bad when necessary. Then Windows & CHKDSK won't see them & won't ever use them. But maybe Acronis does see them.
(2) Partition 5 has an old WinRE.wim inside. Older ones are not supposed to be used. If you want, we can try to move the 2004 WinRE.wim from partition 6 into partition 5. I would move the entire Recovery folder. If you put the letter R back on partition 6, then...
DIR R:\Recovery /a /s
...will show all the folders & files I'd want to move...
Then, we can point Windows to it using ReagentC. Otherwise, I believe Slance310 when he says you may just delete it & let it regenerate when the next Windows version comes in. Actually, maybe that's best. But you'd have to use an external recovery environment until then. That means plugging in your Win Installation USB drive, booting to it, & selecting "Repair your computer" (instead of to install).
(3) I know Shift+Restart didn't work for that computer. You say the other way works...?...
"START, Settings, Update & Security, Recovery, Restart Now (under Advanced Startup)"
That's where you have the "Choose an option" screen. If there, what version does the Command Prompt say it is, at "Troubleshoot > Advanced Options > Command Prompt"? I think it will be 2004, but with a lesser OS Build (which is OK)...
And that will prove the WinRE.wim on partition 6 is the active one.
(4) I think partition 6 should be repurposed to hold Acronis images. We'd delete & recreate it with a letter. Then, see whether Acronis will offer the letter when asked to make an image. But I don't think you'll be able to make sector by sector images. Also -- it isn't as good as putting images on an external drive. If the internal drive crashes, it takes all the partitions with it.
You could have just one/two images on the internal drive for when you go on the road. But bring your Bootable Media with you.