What do the 'additions' from the right panel actually do?

I’m talking about the modifiers from dynamics/tempo - like ‘molo’, ‘poco’, ‘piu’ , etc.

The appearance seems to generally mimic what you can do with prefix/suffix properties on the object, but they don’t set that. So what’s actually going on? Also, is there any way to modify the behavior? Sometimes you need “cresc. molto” and sometimes “molto. cresc.”

Those are a fixed set of common modifiers, the intention being that they should provide a convenient way of getting to the most commonly-needed options. You can’t modify their behaviour beyond the options provided on the Dynamics page of Engraving Options that influences choices over e.g. “subito”. If you need something beyond those modifiers, use the properties.