Yes of course, I can confirm that Hard Disk Sentinel is 100% safe to be used with SSDs.
The goal is exactly to support both hard disk drives, SSDs, hybrid drives, pendrives, memory cards and further devices too - in a single software.
Yes, the tests can be used on SSDs too. Of course if you run a destructive (eg. overwrite) test then as should, it writes the SSD, but such tests are generally not required on an SSD, except if you prefer (for example) to clear all stored data before selling an used SSD.
The generic diagnostics (Disk menu -> Short self test, Extended self test, Read test, Disk Repair test) are 100% safe and does not write anything so does not cause wear.