Would that make sense to format a separate partition with 4096 block size?
Usually yes. It is also the default size used for most filesystems.
Would my IO operations on the files improve? How?
Since you would use the defaults they would not change.
Now if you wanted to format the low level sectors on the disk (and not the way the filesystem on top of those disk blocks is used):
If the disk allows 4KB sector reads you could read 8x more data in a single IO operation. That is not the same as 8x faster, since reading might take longer. It is likely to be somewhat faster though.
Is it healthy and safe for the HDD with otherwise 512 block size partitions?
I suspect that it is neither what you wanted to ask, nor that it is possible with tools which are available to end users.
The closest thing I ever did was low level reformat old IBM mainframe SCSI disks from 520 bytes per sector to 512 bytes per sector. That was the entire disk, including spare sectors.
By now SATA disks use the same trick as those old SCSI disks: When sector reads fails it will try to read the sector a few times. If a re-read works: no big deal. If it happens a few times to the same sector it is assume to fail soon, and a spare sector is used instead. Thi sis all done in the firmware and the end user does not notice this.
Now if you start changing physical sector sizes on the disk then you will also need to rewrite the firmware so it is aware of the different sector sizes and uses the right spares. Technically that is possible, but in praxis you will need factory tools and inside knowledge.