This. If you can sing along to a song, you can definitely figure out tones. It might take some associating each word with a tiny little snippet of music, but that is essentially what it is.
Re: figuring out whether your tones or speech is correct, try talking to Siri or Google voice search. I tried my French on the French Siri and it was humbling.
> This. If you can sing along to a song, you can definitely figure out tones.
I think you understate the difficulty of recognizing and reproducing tones. Quite a few people can't carry a tune to save their life. Even among the majority who passably can, doing so extremely well - aka having "perfect pitch" - is recognized as a rarity.
First of all, that's not what perfect pitch is -- it is rare, but it has absolutely nothing to do with how well you can carry a tune.
But secondly, if the entire population of China can do it, then statistically, you're almost guaranteed to be able to do it too. And the tones in Chinese are really very, very simple. They're no different from the way you end a question with your voice moving upwards, or the recognizable "valley girl" pattern. It's the same idea, just applied to single words instead of whole sentences.
Re: figuring out whether your tones or speech is correct, try talking to Siri or Google voice search. I tried my French on the French Siri and it was humbling.