The terrorist part is complete speculation, although why else would the dad be on the watch list in the first place?
Melvin is also speculation with only hints, but the hints are pretty strong.
Guy in park is different. Drifty in the commentaries all but confirms he's a blood relation of some kind, so whether he's a grandfather, an uncle, or something else, it's a safe bet that he's more than just a random person. The fact that he's being secretly escorted by a government agency, but allowed to wander with a small detail means he's not a flight or rescue threat, so whatever his role is, it's either been accomplished already and he's just under permanent protection, or the threat isn't domestic and he's relatively safe in public in the states.
Everything else you suggested is possible, but doesn't fit the narrative of the story. For Homeland to investigate him for receiving a large sum of money from someone on the watch list means Homeland most likely suspects his dad at least MIGHT have a connection to a terrorist group. The photos that could never be traced were commissioned and leaked by someone two years prior, and the only person that shows up that fits the bill of being interested in him that Homeland shouldn't know about at the time is "grandpa."
I know the narrative isn't perfect, and there are definite logic and timing flaws, but the thing that would make the most sense is that everything IS connected, so I take that as my Cannon.