Adding the required CSS class just triggers jQuery validation. It's not going to do server side validation.

How do I add the icons?

Just add a span element next to your input with the appropriate CSS classes.

I need to look at the code of other elements and just add those spans in the code manually?

Yes.

So what do I need to do, to have server side validation for my input element?

Create an actual fieldtype plugin to handle the display, validation, and storage.

You mean I need to create (by coding it) a new custom plugin, that adds a new field type?

If you want a custom field that has customized display, validation, and storage then yes.

...Something like people have created here and here ?...

No, because those are 8 and 6 year old topics that are no longer relevant. The easiest way to create a new fieldtype plugin is to use an existing one of ours as an example.

In any case I don't need my Custom HTML field displayed and storaged. Only validation, that will check if there's some text in the INPUT element, when pressing the Register button.

Custom HTML fields have no validation. Validation on them isn't even attempted PHP side. This is true for all calculated fields (Code and Query field included) since they have no storage behavior. I guess you could validate the raw POST value using CB Auto Actions, but that's the only thing I can think of to validate your input PHP side.