A few comments:
First, don't panic.
The fact that you've only had your aquarium set up two weeks and you are reading nitrate is a good thing. Google "aquarium nitrogen cycle" for lots of info, but your tank is still cycling. Your turtle excretes ammonia (urea actually, but for the nitrogen cycle it's the same thing). Your turtle is drinking the water it's swimming in. Ammonia is bad, but not to the point where he's going to die with those readings.
In an aquarium, bacteria from the environment start eating that ammonia and converting it to nitrite. Nitrite is bad too, but there other bacteria that convert the nitrite to nitrate. Nitrate is not good, but nowhere near as concerning as ammonia or nitrite.
You have nitrate readings. Unless you have nitrate readings in your tap water, that means you have both bacterial colonies active. That's a good thing. They will take time to catch up, but unless your filter is really undersized, they will catch up.
Once they do, your ammonia and nitrite readings should be zero. There isn't really any appreciable bacteria that will get rid of your nitrate. That's what water changes do.
You're two weeks in and you have nitrate. It took me over six weeks when I started.
I also wouldn't worry about the ph. It is high, but my tap water varies between 7.8 and 8.2. Our turtle is fine.
I would recommend carbon. Is not going to help with ammonia or nitrite, but it will help with other trace elements that are unhealthy. Carbon in the pre-packaged cartridges is expensive, but you can but bulk carbon. I got a $17 bag of bulk carbon this spring that's going to easily last me into next year, and I'm using it for a turtle tank and another fish only aquarium.
EDIT: and in terms of carbon clouding the water, before you put it in the filter, rinse it under the tap until the water flows clear. I've never had an issue with carbon clouding the water after doing that.
