Can manicurists spread the disease Hepa, HIV, Herpes, STD with their cleaning paraphernalia?

Herpes isn't likely to be spread that way or even through sharing / using public toilets. Also stds and HIV are not likely to be spread by using a public toilet, unless their very dirty.
Other infections like HIV or hepatitis are more likely to be spread through re using tools that haven't been sanitized but stds like chlamydia are not likely to be passed out side of having sex.
You can always ask if you can bring your own manicuring kit if you don't want them using their own tools.

Anyone can spread those diseases, even sitting on a toilet seat or infected McDonald's burgers can give you HIV so of course manicurists can spread it.

yes

Hepa Disease

2