Would "I see my grandfather." be an implied translation of the given sentence?
Not exactly. You could see someone else's grandfather, or even en eldery male - in very common speech.
In context, it could indeed mean "I see my grandfather." But as gabe81 rightly says, it could also be someone else's grandfather or even just an old man.
In that case, would you consider "I see my grandfather" an acceptable solution to the original Polish sentence?
In English: "I see grandfather" would imply "I see my grandfather" rather than "I see a grandfather"...
Since Polish has no articles, there is no distinction between "I see grandfather" and "I see a grandfather"--except by context. Although the sentence can indeed in specific contexts mean "I see my grandfather," I would mark that as incorrect (if I had that power, which I don't) because without that context the "my" should not be assumed. Hope that helps!
In English, this more resembles I see Grandfather. Grandfather being the name you use to refer to your grandfather (little g)
My understanding is that this is more: I see [a] grandfather.
So here we have biernik of dziadek which is the same declension as dopełniacz because (dziadek) is singular personal is this correct?
Close. "dziadek" is singular, masculine and animate.
"personal" doesn't matter, because this rule only works for masculine nouns. And not necessarily persons, but also animals.
"dziadkiem" is Instrumental, used in sentences like "You are my grandpa".
"widzieć" (to see) takes a direct object in Accusative, which is "dziadka".