I don't know what you refer to about checking disk integrity. First Aid does not report a hardware problem and DiskWarrior says that the hardware seems ok. Is there a specific Disk integrity check that you are referring to?
--> first aid is probably sufficient. if a disk passes software tests, the hardware is typically ok.
I don't know what you refer to as "Memory Pressure".
--> learn about the new memory management (search my name if you want, on these forums). Apple implemented a cool new memory management approach based on compression. But short answer is go to activity monitor, click on the memory tab, and see if its green. if so, all's ok. If not, get RAM.
Also note, that for a period of time (about six months ago, my computer would just go black and restart on a fairly regular basis - every few days. The Apple Genius Bar checked it out and could not find anything wrong. That problem has reduced to occurring only about once a month.
--> that is just weird and confusing and suggests a larger problem. Hardware? major software issue? Who knows? Run diagnostics (i think its called apple hardware diagnostics or ???)
I saw some note about people using Kapersy security and CleanMyMac. I don't have either to these. I have used MaKeeper, but I think I got it all out.
--> these are all bad actors
Is there any utility worth using?
--> not many. the others you mention (cocktail, malware bytes) are at worst useless - none are harmful. I've played with them. truly not needed though.
Good luck. Sounds like you have a) a potentially major unknown issue, b) maybe some remnants of bad stuff, c) more sleuthing to be done with activity monitor.
Again though - what changed when this began? and separately, have you ensured that all the programs you have are updated for el cap? It breaks a lot of stuff....