A) "I played football with my friends" =good and natural "I played a game of football with my friends" = is the natural way natives would say it "I played football games with my friends" = good grammatically if you played more than one game. "I played a football game with my friends" = good grammatically if you played only one game B) I like watching football games on TV. = good grammatically I watched a football game on TV. = good grammatically I like football games on TV. = natural native way I watched football on TV. = natural short native way. C) also we use the word match for a competition whether professional, serious or just between friends. A) "I played in a football match with my friends" "I was in a football match with my friends" "I played football matches with/against my friends" = good grammatically if you played had/played more than one match. B) I like watching football matches on TV. = good grammatically I watched a football match on TV. = good grammatically I like football matches on TV. = natural native way I watched a football match on TV. = natural short native way.

Yes, "game/s" can be omitted. I played football with my friends. You would not say I watched a football on TV. That would be very odd. But can say that you watched the football on TV.