If I activate the effect of Hand Destruction but I send Kidmodo Dragon to the graveyard, can I activate Kidmodo Dragon's effect?
No you can't.
when you activate a card, you must resolve it's entire effect (or the entire chain if that is the case) before new effects can start activating. (In this case Kidmodo Dragon and Red-eyes will be sent to the grave simultaneously.
So here Kidmodo Dragon's effect will start a new chain after hand destruction has completely resolved.
If at that point you have drawn a new dragon type, you could summon that monster with its effect.