Can't get resolution higher than 1080p on my Samsung UD590 4K monitor

I just got a new monitor and i was trying to get a higher resolution. At first, I was using a Early 2011 MBP, but realized the graphics card just can't do anything over 1080p. My laptop crashed this week, so I bought a new one, a late 15" 2013 MBP, the one with 16GB memory and a graphics card.

I was expecting this new MBP to be able to output 4K, or at least something around 2560x1600. And yes, I have the most recent 10.9.3 updates, which adds more options for 4K output. But higher resolution outputs are not available to output. So far, I have used a MiniDV-to-HDMI cable and MiniDV-to-HDMI-Converter with an HDMI cable. Yet, there's no option to output anything higher than 1080p. It's kind of ridiculous to see how much code fits on my 28" monitor, when compared with my Retina display.

Here's my questions:

1) Are the late 2013 MBP's limited to 4K for specific hardware models? And if so, is the Samsung 28" UD590 included in this list? If not, can support be added?

2) Do I require new cables to get resolutions higher than 1080p? I'm not sure if my HDMI cables are 1.2 or higher. I'm also unsure of the MiniDV Converter I'm using. Do these have to be specific versions? If so, this is pretty confusing. Regardless, I have a MiniDV-to-Displayport-1.2 cable on the way. Will this cable allow me to output anything higher than 1080p? If not, I'm going to be a bit disappointed.

Bonus Question:

Why is it that the Apple Monitors "just work" better? Here's what I mean by this: in the past, when I've plugged my 2011 MBP into an Apple Monitor, the configuration is seemless. Everything just works. With any other brand of monitor, if my laptop goes to sleep for an extended period of time, I will lose my window arrangements. When I restore my laptop from sleep, I have to drag my windows back into place. There's alot of small issues like this that only happen if you use non-Apple monitors. But if you buy everything Apple, you don't experience certain glitches like this. You have a seemless user experience. Why is that?

I've been using a monitor from 2007 (and yes, it outputs 1080p). When it comes to configuration like this, I was expecting my new 2014 4K monitor to be just blissful compared to the one from 2007. However, I'm still experiencing glitches like the one i described above. I'm very disappointed. For example, I have to take about 30-60 seconds to move all my windows back into place (although i don't actually drag them, lol). Can you elaborate?