Well, you can use yoshinator's "half-center" method to improve the centers.
Practice. At your speed, just doing a lot of solves is the best way to get faster.
You should expect to be slower initially when you switch from one method to another. It takes a while to adjust. Maybe you could commit to Hoya or Yau for a while before making a decision? Assuming you haven't already.
Depends on what your weaknesses are. Maybe a video of a typical solve would help?
Well a good method to start with, I think, is reduction but there are many others. It's very simple and easy to learn. You basically turn your 4x4 or reduce it down to a 3x3 and then solve it like a 3x3. Parities are something that can't happen on a 3x3. For example it's possible to get an edge that is flipped the wrong way, two corners switched, or a cross on with only 3 edges oriented correctly instead of two or four. There are both OLL parities and PLL parities. All this means for you though is that you will have to learn some new algs but for the most part, 4x4 is very intuitive.
I don't know about K4 either, but Yau and Hoya are very fast. I would recommend just learning reduction to start with though. They are both pretty much reduction with some different aspects which both have pros and cons. When you learn edge pairing, learn just regular kindv'e freestyle edge pairing wlthat you do one at a time, then when you can get that maybe sub 3:00 average, learn 32223 edge pairing. Once you can get that maybe sub 2:30, learn one of those methods that you asked about. If you choose Yau, 323 edge pairing is really fast, i don't know as much about hoya. For parities, CrazyBadCuber's videos were a good way of explaining what they are, and how to recognize them, but you'll pretty much recognize them to start out with when you are solving the 3x3 stage and say "WTF, that can't happen on a 3x3?" later you'll begin to recognize them much earlier and quicker. Look on the Wiki if you want faster algs than Dan's though. Good luck!
You should really learn reduction first before learning one of those methods. If you don't know how to solve a 4x4 yet then you probably won't understand how to do those methods or at the very least you'll probably struggle with them. Don't worry about speed at this point if that's what you're concerned about. Just worry about learning how to solve a 4x4.Speed will come later. Once you are comfortable with reduction and can solve a 4x4 in less than 2:30 then I recommend switching to a more advanced method. But whenever you switch is up to you. I use Yau with 323, sometimes 62, edge pairing and I think it's pretty fast and it's really easy to learn if you know reduction. Hoya and K4 as well as cage are also popular methods but I don't know a lot about them other than how to execute them.
When doing OLL, if you see 1/3 edges oriented do OLL parity. I use: Rw U2 Rw U2 Rw' U2 Rw U2 Lw' U2 Rw U2 Rw' U2 x' Rw' U2 Rw'. When on PLL if you see a case that you do not see on 3x3 do: r2 U2 r2 Uw2 r2 Uw2. that's a simple version I would watch this video to get more detail, the algorithms are different, but the idea is the same.
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