As a native English speaker, "near to" sounds totally correct, perhaps there are grammar books that claim otherwise. Anyway, we're learning Romanian, not English.
Edit: see here that they are both equivalent: https://dictionary.cambridge.org/grammar/british-grammar/near-and-near-to
yes, both of them are used in the sense of ”close to”, ”near”, and they are mostly interchangeable. However, there are situations when one or the other may be preferred, like in enumerations, or when you want to stress the order (use lângă) or the distance (use aproape de). Ex: you put in order 10 forks on the table, you will say ”le așez una lângă alta”, which would stress the order, enumeration, but if you have only two and want to stress the fact that you put them VERY close to each other, you will say ”una aproape de cealaltă”. Spatially ”lângă” always mean very close (absolute), but ”aproape de” could mean any distance (relative). For example you go to visit a friend who lives 100 km away and he calls you when you are on the way at just 10 km away from his house, and asks you how it is going and how much longer you need, etc, you will reply ”ajung imediat sunt aproape de casa ta” (I will arrive at once, i am close to your house). Using ”lângă” here would mean that you are so close you can touch it with your hand, however, it can be used in figurative sense to stress that you are ”really close”.
Yes, if I hear ”Danemarca nu este lângă Grecia”, I understand without doubt that they are not neighbors, they do not share borders. But this may include a bit of subjectivity too, as I know where the countries are :). Rhetorical, you can say the same sentence to stress the fact they are far apart. Like ”the mosquito is not an elephant”, or ”iarna nu-i ca vara” (the last one a famous quote of the former Romanian president).