Is my HDD failing or not? (false information?)

Ok, first and foremost, back up that drive. If you can, back it up to 2 separate drives. Do not trust it as the sole repository for any important data whatsoever.

Your pic shows "There are 33360 bad sectors on the disk surface. The contents of these sectors were moved to the spare area." Every disk has some bad sectors to start out with, which get marked at the factory and go unused for the life of a disk. Then there's some spare area set aside in case a couple sectors go bad later on, so it can remap them and keep chugging. That's why your scan doesn't show any bad sectors, once they were found the first time they got marked bad and remapped to the spares.

A couple sectors going bad isn't necessarily cause for alarm. 33360 sectors going bad is roughly 33000 more than the point I would start panicking, rounding down to the nearest thousand. People usually start to notice something wrong when it reaches low double digits, like say, 12.

You can look at the SMART tab for more information about what Reallocated Sector Count means and the specific numbers. But, do that after making backups. That will fail, very soon.

Ignore failure warnings at your own risk. as TMTOWTSAC said, backup the data FIRST, then worry about additional testing. The photo you posted shows that the disk is too close to failing to risk anything else.

Short answer: YES!
Long answer: The issue here is the re-allocated sectors.There are only so many "spare" sectors that can be used for this purpose. Once they are gone, and the drive needs to re-allocate another sector, and can't, the drive has failed, and you will have to spend big bucks to recover any data from the drive.

I'm not an expert an interpreting SMART info, but here goes. The Reallocated Sector Count keeps track of how many sectors have gone bad and been mapped over to spare sections of the drive. Those spares are set aside at the factory and there's a limited amount. The values you see are generally normalized on a scale, rather than the actual numbers. Just like saying "45%" doesn't actually mean "45", rather, it represents a number to be derived from another number.

In your case, the first value Hard Disk Sentinel shows is the Threshold (Thre) column. The Threshold is the value below which the drive is considered to be failing. In this case, 36. But that doesn't mean 36 reallocated sectors. It's a score of 36 on a normalized scale representing some number of bad sectors on their scale. For the Seagate ST500DM002, the scale starts at 200 (had to look that up), which would mean the total number of reallocated sectors does not equal 1 increment. Once you reach that amount, it would go down a point to 199, and so on. The points on the scale were determined by the drive manufacturer based on their drive failure data.

The actual number of bad sectors at each point on their scale depends on the drive of course. Larger capacity drives have more total sectors, so they would have larger increments. Enterprise drives are designed for higher reliability, so they would have smaller increments than the same capacity in a consumer drive.

Your count is at 1, which is below the Threshold of 36. It's way below, far worse than just 35 lower. It's at the very bottom of the scale and can't go any lower. It may have reached that point some time ago and kept getting worse. The value at the very end is the raw data, the actual number of sectors, given in hexadecimal (base 16.) 0000000000008250, which = 33360 in base ten.

Anyway, 33360 is...it's a lot. It's more than a lot. A couple sectors going bad over time is normal and doesn't necessarily mean much. But 33360 means something is very wrong, something that is probably continuing to damage the disk. Like moisture contamination and the platters are rusting. Or the magnetic layer is defective and losing the ability to hold a charge. Be thankful you got some warning and a chance to back up your data, that old Google study had 36% of all HD failures give no warning beforehand.

Back up your data, RMA the drive if it's under warranty, and get a replacement.