Can a card cast using its flashback ability be countered by a spell such as Cancel?
Yes. The only thing that flashback modifies is where the spell is being cast from when paying it's Flashback cost before being put on the stack, and where it goes when it leaves the stack if its Flashback cost was paid.
702.31a Flashback appears on some instants and sorceries. It represents two static abilities: one that functions while the card is in a player’s graveyard and the other that functions while the card is on the stack. “Flashback [cost]” means “You may cast this card from your graveyard by paying [cost] rather than paying its mana cost” and “If the flashback cost was paid, exile this card instead of putting it anywhere else any time it would leave the stack.” Casting a spell using its flashback ability follows the rules for paying alternative costs in rules 601.2b and 601.2e–g.