Does my client need to be running to "mint" new NXT coins?
It depends on what you call "client".
That original funny web interface makes things a bit complex in terms if "client", "server", "peer", "account".
I prefer to name original software (Jetty + Nxt servlet, that you see as terrible black window) as "server". So we have p2p network of servers.
Browser, connected to one of servers, acts like lightweight "client". Client is dumb, it only show data from server, accept clicks on buttons and so on, but it doesn't perform any actual work for Nxt net.
Actually, there can be different client software even for original server, because server provides some API. And it can be not only browser. But for now there's only one original client.
"Account" is just a piece of data inside server's memory. When you unlock account, server software just create some bytes in memory and start to process it.
So, answer is: you need to unlock your account on some server (better you own server), than you can close your browser - server doesn't need it to perform work for net, and for mining too. If you restart server, you need to unlock account again.