Are some of the tones in audio clips incorrect? For example, in "greeting two", when it asks "what sound does this make" for "xìng", the audio sounds very clearly first tone when it is meant to be fourth, but when it plays the audio for the full word "gaoxìng," it sounds correctly fourth tone. This happens with a few other syllables, too, where the audio plays the wrong tone when presented alone but correctly in context.
Has anybody else noticed this, or am I just really bad at hearing tones?
You're right - the pronunciation is wrong. It should be fourth tone for the 兴 in 高兴, so the pinyin is correct.
But I don't think it's as embarrassing a mistake as it seems (still not acceptable!). 兴 can be pronounced with a first tone or fourth tone - the meaning depends on which one you use. With the fourth tone it means excitement, and the first it means to prosper (among other things). It's a bit like two different words that use the same character. There are a few characters like that, for example, 了 can be pronounced "le4" or "liao3".