Did you try to turn them in in the same system where you got the kills?
If you score a bounty it will pop a big message in the top right of the screen showing how much that bounty was worth. To claim it you must dock at a station in that system or somewhere else with the same controlling faction. Outside of these two cases you can use an Interstellar Factor which can be found in low security systems and they take a 25% cut.
You can also see if you have any bounty claims in the transactions tab of the left panel.
I don't think the last 15 seconds thing is true. I've gotten bounties more than a minute after ceasing fire and switching to a different target as the NPCs finish him off. But to be honest I'm not sure how it works really. Just shoot as many as you can and it will add up quick!
Yeah I did - for both I definately got shots on target, I do know that, but I can't swear to killer blows, certainly not the first NPC but not a bad volley of fire on the second... but never mind...
What I DID learn is that my gimballed beam lasers were rubbish and that my multi-cannons (also gimballed) were far more effective... so some changes next time I take the Viper out for some shooty bang bang.
Embarrassed to say I also learned that all the people on the left of the mission board can offer you missions and not just the first name on the list! Oops!
Gadded about all over the place doing a bunch of missions in the Viper... got some bounty kills in the morning... so all in all a nice day ED-ing....