I'm a German native speaker doing the placement test for fun. Here it sounded kind of awkward to me: "... a lot of high trees". Do you really say that in English? I was rather thinking of many trees. Also, "Dieser Wald" would certainly be "These woods", its the whole area consisting of many trees/lots of trees ;)
Yes people would say "many" or "a lot of" or "lots of" fairly interchangeably. However I think they are more likely to say "tall trees" than "high trees". "High trees" is what makes your sentence sound a little awkward to me. "Tall" sounds better because it refers to stature extending from ground/current level to a higher level. "high" just refers more to an elevated position rather than stature, although some people will use the terms interchangeably. The difference between the two is more obvious when talking about people, where we refer to a person being tall, but never as being high when referring to their stature.