What influenced Tertullian's view of God's corpus? Why did he believe that God (although he is spirit) has a body? Firstly, Tertullian's belief is basically informed by Stoicism but I don't think that it is only Greek philosophy which shapes his view of universal corporeity (i.e. the doctrine which asserts that every existent thing has a body). His view may have been influenced by the notion that existing things must have some kind of tangible substance in order to exist.
It is also true that Tertullian is possibly the only early theologian to argue that God has a body of any kind. I've read that Melito of Sardis affirmed the same doctrine. However, I have yet to come across an explicit mention of this doctrine in the writings of Melito. The early writers of the church tended to view God as incorporeal or bodiless: Origen is explicit on this matter in De Principiis. However, Lactantius and Novatian imply that God is corporeal. But they also were influenced by Stocism.
The Witness position on God's spiritual body is basically inferential. The Bible never explicitly says that God has a body. It does talk about a "spiritual body" in 1 Cor 15:42ff and what the expression in that chapter possibly signifies is a matter of debate.
Witnesses basically reason that in order to see God (1 John 3:1-3) or behold his presence (i.e. face), a divine spiritual body must exist (Hebrews 9:24; Rev 22:5). God must be corporeal in some sense. I guess that one could also reason that since Christ has a body and he is the image of God, then the Father likely has a body too (1 Cor 15:45-49) I ultimately believe the best that one can do is to make a circumstantial case for God's spiritual body since the Bible is silent on the matter. It does not discuss God's body or state that he has one which is not to say that I'm denying God's actual or possible corporeality. I'm just contending that we might have to accept the limits of what can be known about this subject and be content therewith.
Louw and Nida's Greek-English Lexicon says the following about the term PNEUMATIKOS in 1 Cor 15:44:
"pertaining to not being physical-'not physical, not material, spiritual.'" This resource adds the following observation: "In some languages the concept of 'spiritual body' can only be expressed negatively as 'the body will not have flesh and bones' or 'the body will not be a regular body'" (semantic domain 79.3).