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, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
If you mean writing the Name of Allaah by itself, we do not know of any basis (Sharee'ah evidence) for doing that. But if you mean writing the Name of Allaah with a Quranic verse or supplication for the purpose of seeking treatment, then we do not see harm in that on condition that the Name of Allaah is written with a ritually pure substance, on a ritually pure object, and in an appropriate place.
The Hanbali scholar Ibn Muflih wrote, "Shaykh Taqiyy-ud-Deen Ibn Taymiyyah used to write on the forehead of the person suffering from nosebleed the verse that reads (what means): {And it was said, "O earth, swallow your water, and O sky, withhold [your rain]." And the water subsided, and the matter was accomplished} [Quran 11:44] He said: 'It is impermissible to write it with blood like the ignorant do because blood is ritually impure; so it is impermissible to write the words of Allaah with it." [Al-Aadaab Ash-Shar‘iyyah]
Ibn Taymiyyah also wrote, "The Quran should be honored and revered wherever it is written; it is impermissible to write the Quran where it would be disrespected."
The principle evidence in this regard is the report narrated on the authority of ‘Awf ibn Maalik who said, "We used to recite Ruqyah in the pre-Islamic days so we said, 'O Messenger of Allaah, what do you say about it?' He, sallallaahu ‘alayhi wa sallam, said: 'Let me hear your Ruqyahs (so that I can judge them). There is no harm in the Ruqyah which does not comprise idolatry.'" [Muslim and Abu Daawood]
Ash-Shawkaani wrote, "This Hadeeth indicates the permissibility of using Ruqyah and medication with anything that does not involve harm or prohibitions in the religion." [Nayl Al-Awtaar]
As for your second question, we have underlined in Fataawa 87157 and 90105 that shaving the pubic hair and the underarms are among the Fitrah traditions and that it is disliked for a Muslim to neglect these.
Allaah Knows best.