Some people like to see the "smoking" effect of a hot compost pile but the reality is that, even if you don't see that, your compost pile is still working. It just may take longer. That doesn't mean you have to leave it all dry or all wet, though.

Your description of the moisture level sounds about right. You can keep the ratio anywhere from 4:1 up to 30:1 for browns to greens. There is a large leeway because compost happens. People don't get it exactly right based on some book-learning, but they still end up with compost. There are people who will mulch leaves and others who don't, and they all still get compost.

When the compost is a good mix of crumbly, dark stuff that you don't recognize what it was before, then you can say that part is done (though some people would at least give it a few months or more to get there). There may be other sections of the compost pile that are not done, so those you can leave in longer if you have just 1 pile or put them to another pile to redo.

You don't need to cover the compost pile. If draining and aeration is good in the pile, then rain should be fine.

If you know the stuff that went into your compost, then it should be safe for your plants. It may still not have all the different levels of trace resources that plants need, but that is where you will learn more and add in variety of materials or mix with other additives when you are ready to use it.

1) Moist is ideal.

2) Cover is not absolutely necessary, but if the compost gets saturated it will slow it down due to less air getting in. It can also leach away a fraction of the nutrients. I use a cover when I can, if I goof it's not the end of the world.

3) If you don't get compost but still have browns left and it doesn't seem to be breaking down anymore after a few months, the pile could use more greens. If it gets smelly and gooshy, too much greens, add browns.

4) Finished compost can't really hurt your plants so there is generally no need to test. I would test the soil you're starting with. After a few years adding it to your garden, your nutrient levels will change, so it's good to have another soil test.

Re: turning and using a 3-bin system. 2 weeks is fairly aggressive although it's OK especially if you have an ideal green brown mix. You can get compost fast that way. What I do with a 3-bin is to always put fresh material in bin 1. When you use finished compost out of bin 3, and bin 1 gets full, turn 2 in to 3 and 1 into 2 and start over. Each batch gets turned twice. I find I always have a batch ready to use (or nearly so) that way, it probably takes longer, but it's a lot less turning. There's no rule though so do whatever works for you!

Great, thanks for the advice. I'm thinking of posting a video later this week so I can get some feedback on the actual composition of my compost.

BTW, the reason I turn the piles every 2 weeks is so that the wetter compost gets a chance to dry out a little by being moved to the top of the pile, and vice versa.

It's not at all hard to test your compost. You'll need two flowerpots. One with regular soil, and one with the same sort of soil mixed with a lot of your compost. Try to sprout carrots or radish in both pots and see if the pot with the compost sprouts as quickly as the plain soil.

When compost is not 'finished' it contains acids that inhibit germination and root growth.

1. Just moist. Excess water will exclude the air the bacteria that will be digesting the material also need to function and they will not.
2. I would and did. Rain can easily saturate that compost and it could then take months for the material to dry out enough to be digested. Remember water displaces air and that could lead to anaerobic digestion.
3. That can be difficult since everything has some Carbon and some Nitrogen. The C:N ratio of tree leaves (most think of them as browns) will vary from 40:1 to 80:1 and sometimes a pile of fresh tree leaves will digest just fine withou7t the addition of an N source while other times they will need that N. If when you put a mix together and it is not heating up as desired remix it with some more N.
Anytime a compost pile freezes solid that says there is too much moisture in the mix.
4. The nutrients in compost are not very soluble and therefore do not show up much in the soil tests used today. As Tox indicated finished compost will not do any harm to the garden. It is easy to tell when compost is finished by appearance and smell. Finished compost will not have much, identifiable, of the original stuff and it will smell of good rich earth, pleasant.
Perhaps this Composting Tutorial will be of some help.

Here is a link that might be useful: Composting Tutorial