Is the method used for istinja' (cleaning oneself after urinating) correct? If not, what is the correct method? And will Allah forgive the lack of proper purification due to ignorance throughout the past years?

Wiping with tissues is not obligatory, and limiting oneself to water alone is sufficient for removing impurity. This is better than limiting oneself to stones or similar items. If the person cleaning himself after relieving nature (al-mustanji) uses only water, he must wash the place of impurity until he is certain or reasonably sure that the impurity has been removed and the area has returned to its rough texture. There is no specific number of washes required; rather, the aim is to be reasonably sure that the impurity has been removed.