Smartctl retest bad sectors

There are no reallocated sectors because they failed to reallocate. Your drive is showing 5 Offline_Uncorrectable sectors, which happens when automatic repair fails. There are obvious read failures shown in the dmesg output, SMART errors, and read failures from SMART tests. There are ways of repairing these sectors as you have mentioned in the question, but from my experience it is a very short-term fix.

Replace the drive.

Your disk had some problems with reading data from the surface, but it seems that the disk dealt with it. I had similar situation:

Error 29 occurred at disk power-on lifetime: 18836 hours (784 days + 20 hours)
  When the command that caused the error occurred, the device was active or idle.

  After command completion occurred, registers were:
  ER ST SC SN CL CH DH
  -- -- -- -- -- -- --
  40 51 08 00 40 37 e6  Error: UNC 8 sectors at LBA = 0x06374000 = 104284160

  Commands leading to the command that caused the error were:
  CR FR SC SN CL CH DH DC   Powered_Up_Time  Command/Feature_Name
  -- -- -- -- -- -- -- --  ----------------  --------------------
  c8 00 08 00 40 37 e6 08      03:39:32.447  READ DMA
  c8 00 08 f8 3f 37 e6 08      03:39:32.447  READ DMA
  c8 00 08 f0 3f 37 e6 08      03:39:32.447  READ DMA
  c8 00 08 e8 3f 37 e6 08      03:39:32.447  READ DMA
  c8 00 08 e0 3f 37 e6 08      03:39:32.447  READ DMA

And when I wanted to perform test, I got:

Num  Test_Description    Status                  Remaining  LifeTime(hours)  LBA_of_first_error
# 7  Short offline       Completed: read failure       90%     18845         104284160

Ultimately, I managed to unblock the sectors, and after running the extended test, which scan the whole surface, I got the following result:

Num  Test_Description    Status                  Remaining  LifeTime(hours)  LBA_of_first_error
# 3  Extended offline    Completed without error       00%     18858         -

If there were bad blocks, they could be observed in the table under:

  5 Reallocated_Sector_Ct   0x0033   200   200   140    Pre-fail  Always       -       0
196 Reallocated_Event_Count 0x0032   200   200   000    Old_age   Always       -       0
197 Current_Pending_Sector  0x0032   200   200   000    Old_age   Always       -       0

In your case, there's no indication of bad sectors because the extended test was performed (11746 h) after the last error occurred (11706 h). So, you can sleep peacefully. :)

As I mentioned in comments, there's two types of badblocks. Here's short info about the difference between the two:

There are two types of bad sectors — often divided into “physical” and “logical” bad sectors or “hard” and “soft” bad sectors.

A physical — or hard — bad sector is a cluster of storage on the hard drive that’s physically damaged. The hard drive’s head may have touched that part of the hard drive and damaged it, some dust may have settled on that sector and ruined it, a solid-state drive’s flash memory cell may have worn out, or the hard drive may have had other defects or wear issues that caused the sector to become physically damaged. This type of sector cannot be repaired.

A logical — or soft — bad sector is a cluster of storage on the hard drive that appears to not be working properly. The operating system may have tried to read data on the hard drive from this sector and found that the error-correcting code (ECC) didn’t match the contents of the sector, which suggests that something is wrong. These may be marked as bad sectors, but can be repaired by overwriting the drive with zeros — or, in the old days, performing a low-level format. Windows’ Disk Check tool can also repair such bad sectors.