I would say that you should focus on accuracy, first and foremost. Speed will come with practice, but it's very important to maintain that 99%+ accuracy.
I've started with an average around 50 wpm, with an average accuracy of 95 or so. Now my accuracy is usually around 98 or higher (getting that 100% more often every day) and my average speed has increased to around 70 wpm.
So yes, I'd say you can have both for sure. Focusing on accuracy will naturally increase your speed as you'll spend less time correcting your mistakes and more time typing.

I would say that there is a way. But you should personalize this to what you think the best way you learn is. I started off with an average of around 105-107 WPM and an accuracy of 97.07% and I currently have an average of 131 WPM and an accuracy of 99.01% It takes work, but Accuracy > WPM for sure.

I would say a good idea is to prioritize getting better at both, but in blocks. Work on accuracy for a while, and don't care too much about the speed. Then focus on speed, without worrying too much about accuracy. Switch focus between them when you plateau on your current focus.

Actually, accuracy is definitely more important. No matter how your speed is low (-50WPM or so), you can duplicate your speed as long as you can control your accuracy. I'm almost at 160WPM average for the last games and I mostly get 100% when I type at 150/160. At this level (150+), slowing down and typing accurately is very painful and difficult. It can be even impossible for certain cases. In contrast, sprinting carelessly takes less concentration and can lead to a decent score like 200. However, I find it pathetic to rely on lucky keystrokes to have a score that we can't naturally get. Anyway, the faster you'll become, the more you'll understand that speed is priceless without a good accuracy.

Eh, it depends on the context.

I argue that some people take the quest for 'accuracy' too far, at a great detriment to speed. As somebody else pointed out, I think sometimes you have to loosen the reins a bit and allow yourself to make a few mistakes, to see how fast you 'could' be typing. As well as being quite motivational, it allows you to have a 'spread' for your scores. You can tell yourself 'ok, I can type 75wpm with 99% accuracy, but 90wpm if I allow only 95% accuracy'. This can be quite useful to know. There are many situations where you don't need perfect accuracy when typing. For small errors---hello, that is what spell check is for. If the difference between 99% and 100% accuracy is a rather large difference in wpm, then I think there is something wrong with that.

Try typing sometimes for accuracy, and sometimes for speed. And then, knowing how fast you 'can' type while allowing errors, start to take away those errors, until you can now type at 90wpm with 99% or better accuracy. Then try pushing yourself again to reach new heights, again, allowing yourself some margin of error.

I personally did a lot of my typing as a kid playing games. Did I worry about letting through a few spelling mistakes now and then? Not a lot. In most contexts I was simply pushing, always, for more and more speed. Over time, I reined in my speed and fixed those spelling errors and typos, and now I can both type with high speed and high accuracy, depending on what the situation demands.

Though I would be very interested to see if there are any scientific studies which can confirm which approach works better to learn to increase speed of typing the fastest: focusing on speed then correcting errors, or focusing on perfection and then increasing speed? I have only my own experience to go on, but I would suspect that a mixture of both approaches would yield the best result overall.

Other factors can influence typing speed that a lot of people don't really think about.

-the complexity of the text being typed
-the language of the text
-delivery method of the text (reading text off a computer screen or typing what others are saying? Is the text easy or difficult to read (in terms of size/color/type of font, etc))
-mental condition of the person (are they alert or distracted, are they stressed or under pressure)
-physical condition of the person (are their fingers warm and well-rested, are their eyes well-rested, are they sitting with good posture)
-how long the typing session lasts (a couple minutes, or hours?---maybe you can burst 150wpm by focusing really hard for 2 minutes, but you can only sustain 100 when typing for more than 5 minutes; do we say that you have 150 wpm, or 100wpm?)
-condition of the keyboard (high quality mechanical or budget, worn-out keyboard)

Yeah anyways, my personal approach is to let accuracy slip a bit when I really want to type as fast as possible, and accept that afterwards, I may have to clean up one or two errors in a long sentence. In many informal contexts like gaming, the errors don't bother anybody anyways, and so I can deliver my message and have it understood with maximum speed. In other contexts, like a research paper, I still don't care about a few errors---the paper will be double and triple-checked before submission, so typing errors will be picked up anyways. Only in situations where I need to type the right thing the first time, because I only get one shot at it---which is not very many situations for me---do I really try and get 100% accuracy, and usually I can't sustain it anyways, so why bother? I find for the occasional error, it is faster to just correct it as I go than it is to slow down my typing speed enough to prevent any errors from popping out in the first place.

I would say type as comfortable as you used to, accuracy and speed come with practice.

At least in the context of the typing tests on this site, better accuracy leads to better speed. Nothing slows you down more than spamming the backspace key.