Yep looks like you got him good. :thumbup: Missile trails are not supposed to go all the way to the target, as their motor only burns for the first part of the flight. The rest of the way they fly with the gained inertia (motor off, so no smoke trails).
Your victim looked like an Su-25 or maybe A-10 to me (rather straight wings), but it's kinda hard to tell on my monitor. In the future when you are in doubt simply press the Tab key to open the chat window. You should see a kill message there, something like [Player1] with [Aircraft1] killed [Player2] in [Aircraft2].
Hope that helps. Welcome to DCS. :)