“...finger has been injured for about a week now.” (1)
This means that an injury occurred to the finger roughly seven days in the past and it's still injured. The same meaning would be conveyed without “now” at the end.
“...finger was injured for about a week now.” (2)
Your first guess is correct. This is ungrammatical unless it's some sort of folksy way of calling attention to something in the past that's about to be the foundation of another point (like “Now listen here...”).
“...finger was injured for about a week.” (3)
Without “now”, the statement is grammatical again. All we can conclude from this version of the sentence without further context is that a finger injury occurred at some point in the past and healed seven days later (the finger is now healed). This statement would probably go with another that would provide greater detail about when the injury occurred, or when it healed (like “The dog bit my hand on Easter.”).