I would need know whether it is correct to say that an idol can be nullified by any idolater without Minhag differences even if he/she did not worship it. And is it correct to say that objects used to contain or hang an idol or even to cover it without giving a religious devotion meaning to the act of covering, are not forbidden to Jews of any Minhag if they were not used in order to honour the false god (ex.: clothes worn by people believing to be gods, nail used to hang an idol etc.)?

Yes, any idolater may nullify an idol even if he/she did not worship that idol. It is difficult to guarantee that there is no Minhag that is Machmir, but the basic rule is that objects are forbidden only if they are used to beautify or adorn or glorify the idol.