All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.
It is apparent from the question, the questioner does not distinguish between 'O, Most Merciful' (Ya Rahman), and 'The Most-Merciful', and condemns those who write it down with definite article 'The' and wants them to write it down with the preposition 'Ya' 'O, Most Merciful' in all cases.
There is a difference between 'O, Most Merciful' and the expression 'The Most-Merciful'. The first expression 'Ya Rahman', ('O, Most Merciful') is a supplication: 'Ya' in Arabic is used to call upon or supplicate, while 'Al' (The) is used as a definite preposition. So we should not condemn whoever writes the definite preposition 'The' when writing 'The Most Merciful' if it is not intended to supplicate.
The same applies to all the beautiful names of Allaah. If we make a supplication we use 'Ya'; when it is just meant to mention the Name, then the preposition 'The' is used.
As regards the second question, which is naming a person after Allaah's names, this is of two kinds:
A. Permissible: To name after Allaah's names; names that could have a meaning common to Allaah and humans, like: 'Al-'Azeez' (The All-Mighty), 'Al-Kareem' (The Most-Generous), and the like, whilst believing that the name and attribute when referred to Allaah is in a way that befits His Majesty but when referred to humans, they are used in a way that suits them. Therefore, one can be named 'Azeez', for example.
B. Not permissible: To name after Allaah's names and attributes, names whose meanings Allaah and humans do not share, but rather absolutely exclusive to Allaah, The Most-High, like 'Al-Khaaliq' (The Creator), 'Al-Razzaaq' (The Provider of sustenance), 'Al-Muh'yee' (The One who gives life), 'Al-Mumeet' (The One causes death) and so on. Therefore, one may not be named Khaaliq, for example.
It is also not permissible to name after Allaah's names in common names using the preposition 'Al' (the) like: Al-'Azeez, Ar-Raheem, Al-Kareem and the like, except if the name is given to describe a quality of the person, not as a name, in which case it is permissible, as Allaah says (interpretation of the meaning): {'The wife of al-’Azeez said:…}[12:51].
Allaah knows best.