I'm not explaining it clearly enough I guess. QS is really not about the landing page per se. Having the keywords in the domain or URL does nothing and is not a factor.
Here's the lowdown. QS, at least in Adwords, but very similar in Bing and others, is composed of about 65% CTR, 25% keyword relevancy and 10% "other". Of this other factors, most if not all of it is page loading time. The keyword relevancy, unless you got so little text the system can't figure it out, should not be a problem. It will be a problem only if your page is about cat food and you use dog food as a keyword. This would make your ad irrelevant anyway and people wouldn't click on it. You also don't need to have each and every keyword on the page. Google is smart, it knows "feline nourishment" is equivalent to "cat food". It's called latency.
So if you want to say that keyword relevancy is a landing page component, fine. That's 25% of the score plus the 10% for loading time and probably a few other small things. Total is 35%, give or take, that's one third of the score.
I'm sure you can nail that one third. There's no reason you can't. If you have irrelevant keywords however, you just lost most of it. But it doesn't mean you'll be fine on the rest (the two thirds). That's all CTR. You may think your CTR is high but that doesn't mean anything because you have no way to compare to others. Actually, that's what your QS is doing. So if your QS is 4, while the average QS would be 5, it means you are doing worse than average. Doesn't matter what your actual CTR is which depends on positioning anyway.
QS is calculated taking position into account and compares your CTR there against what others have done with that keyword. Again, if your QS is 4, it means you are below average. You can have any QS for any CTR. I've seen average QS for what I thought was high CTR myself, often 10% or more in the top spots. It was only telling me that others have done much better. I've also seen QS of 10 in top positions for low CTR, a few times below 1%. It was telling me that my CTR was above average, which means others were not doing any better. It's all relative.
A QS of 1 is the off switch and usually means you're violating a rule.
Using all three (actually four) match types in a group is perfectly fine. No need to split hairs and have only one keyword per group. Besides, Adwords treats all keywords in different match types as the same. So "dog food" = [dog food] and the QS will always be the same for those. Of course, if there's a good reason to split into a separate group, the QS will be different.
Don't kill under-performing keywords. Find out why they are. Maybe you need to tighten your group. Groups should be themed. Therefore, you may have a general dog food group and others themed around small dogs, big dogs, diabetic dogs.
The cutoff should not be impressions but clicks. Test two ads only at one time. When they both get 20 or more clicks, replace the underachieving one with a new test ad. Rinse and repeat. With 20 clicks each ad, you have about a 90% certainty that the better one (higher CTR) will consistently outperform the other. The more clicks, the higher the certainty. That's what is meant by give time for the dust to settle.
Never test more than two ads against each other at a time. Three if you have lots of searches each day, talking thousands here, which doesn't happen often.
I too hope there is some method to get the QS higher you know we could make a decent amount if that was the case. 