"I thank my grandfather" was not accepted even though "I thank grandfather" was one of Duo's accepted translations.
Based on one of my old German grammar books substituting the definite article for the possessive adjective is preferred when referring to persons or items in one's private sphere. Thus one says "Ich frage den Vater" rather than "Ich frage meinen Vater" when one wishes to say "I am asking my father." Duo does not appear to accept translations like this. Perhaps I am out of date on this matter.
No, that would be ungrammatical and unnatural in English. You could say "I thank the grandpa," but adding the word "to" before the direct object doesn't work here. If you really want to use the word "to," then you could also say "I give thanks to the grandfather," although that construction is a bit more formal and is less common, especially in everyday speech.
Großvater is a masculine noun that is dative in this case, and the correct article for masculine dative nouns is DEM. DEM is also used for neuter dative nouns.
DEN, meanwhile, is used for masculine accusative.
Here's a chart that I think might help:
I've also found that writing the chart out a few times on your own can be very helpful.