Build lots of ap.

Learn the exact range of the AoE on Lay Waste - knowing if you can get the single target damage helps a lot with last hitting and bullying in lane.

A bit of both. If you want to get a Requiem off, you have to have at least the 3 seconds needed for the channel left in Death Defied (it will lock you out otherwise), but you can usually get ~3 Qs off first. You don't want to be just mindlessly mashing R every time you die, though. Only use it if it will actually help. (And try to avoid random KSs with it. If you don't need to ult to secure the kill, don't.)

Best way to avoid the KS is to keep in communication with your team in those situation. There had been cases where my partner and I missed our kill and we just called my other friend who was playing mid karthus and he ultied and killed the guy for us. If you used your ult to prevent the mid from escaping and you happen to have killed someone else as well and say get a double on accident I am pretty sure that your teammates will understand that you meant to use it on your target and not theirs.

The great thing about Karthus' ult is that you can literally hit the enemy anywhere, the down side you can also steal your teammates kills. So just be mindful and stay in constant communication. Also to avoid ganks I do recommend to get some wards and ward the bushes if possible especially against junglers. that can definitely give you an earlier heads up on incoming ganks.