Bran Stark has a strength number but no challenfe symbols. How do I know what challenges he can participate in? If he can't participate in any then what is the point to having a strength number?

He can participate in all the challenges that he has icons for -- so none as printed.

But he can gain icons through effects, so he might need a STR number at some point. He could also have STR modifiers and effects that kill or discard him at 0 STR (like the plot Threat from the North), so he might need a STR number at some point. But in the end, he's a character card and all characters have a STR number, even if it is 0. Characters also have from 0 to 3 challenge icons. He's an example of one that has 0. There are others (though not in the Core Set, I think).

If Bran is Standing, his Strength counts toward Dominance.

Characters also have from 0 to 3 challenge icons. He's an example of one that has 0. There are others (though not in the Core Set, I think).

There is Maester Aemon as well.

There are also effets that give icons to characters. In the Core Set, there is Summer and Lordship.