Q. 1. Does the Catholic Church or the Holy Bible say anything about marking one's body with tattoos?
A. 1. I am not aware of any known Catholic teachings on the matter of tattoos. I can only assume that the Church leaves it to individual Catholics to decide on this matter.
At the same time, the Holy Bible does say something on the matter of tattoos. It states: "You shall not make any gashes in your flesh for the dead or tattoo any marks upon you: I am the Lord." [Leviticus 19:28]
The biblical prohibition of tattoos was intended to separate God's people from the pagan society that used these as religious symbols. The usage of tattoos in the form of images or numbers has continued to this age, such being practiced by the Nazi, street gangs, bicycle gangs, prisoners, as a sign of membership, etc... Most of these associations are non-christian. While it is not necessarily so in every case, when one sees a tattoo on a person, it is assumed to be associated with evil or immorality (in the sense of crime, drugs, alcohol, theft, bullying, open sexuality, etc...).
Your body is God's gift to you; what you do with it is your gift to God. If you smear it with images of serpents, naked bodies, weapons piercing the heart (killing), demonic figures, then God shall surely have the last word on the matter.