Never Gonna Get It

Answer: A clock.  "Not a bird but I can fly through time": This is a metaphorical use of "fly". A clock doesn't physically move through the air, but the passage of time is often described as "flying" (as in the saying, "time flies").  "Hands moving, moments chime": A clock typically has "hands" that move to indicate the time, and some clocks "chime" to mark the hour or specific intervals. The riddle uses personification and metaphor to describe a clock's function in a creative way.

Answer: Mine.   “Mine” carries three distinct meanings that match each clue: Shouted to claim: someone yells “Mine!” to assert ownership. Dug for riches: a mine is a place where minerals, coal, or gems are extracted. Buried to maim: a land mine is an explosive device concealed underground. The riddle packs these homonyms into one word, making “mine” the perfect answer.

Answer: Ice Cream.  Here's a breakdown of the clues: "I am sweet and cold": This directly describes the primary qualities of ice cream. "with a stick to hold": This specifies a common way ice cream is served as a convenient handheld treat, often with a wooden stick. "a treat on a hot day, worth more than gold": This highlights the refreshing and highly desirable nature of ice cream when the weather is hot. 

Answer: A Mailbox. A “mailbox” has “letters,” but they’re mail, not alphabet characters. It also has a “flag,” the small lever you raise to signal outgoing mail, even though it isn’t a pole with a flag. So both clues fit a mailbox through wordplay.