How do you pronounce Qatar?

Everyone does that with place names and personal names, in cases where:
  • The speaker's language simply doesn't have the same phonemes as the original. E.g, Göte, Dvořák;
  • The original language presents consonant combinations to which the speaker is unaccustomed. E.g, Plzeñ, Hranice (yes, I'm picking on my mother's language a lot but there's probably no language with more difficult phonetics for anglophones than Czech); or
  • Transcribing foreign writing into the Roman alphabet is misleading and/or ambiguous, e.g...
My third example. Mandarin J is not English J. Mandarin has no voiced stops or affricates. The English paradigm of voiced/voiceless consonants does not match the Mandarin paradigm of aspirated/unaspirated. The consonants in judge, ads, big are voiced, whereas the consonants in church, eats, pick are voiceless. In Mandarin the consonants in ping, cha, cai (pronounced tsai), qi (pronounced chi) are voiceless and aspirated, whereas the consonants in bing, zha (pronounced ja, more or less), zai (pronounced dzai, more or less), ji are voiceless and unaspirated.

So the J in bei jing (northern capital) is not the J in jingle. It's voiceless. It's hard to describe because it never occurs in English. If you consider that Mandarin T and D are like the T in English top and mistake, respectively, that gives you a clue to Mandarin J. But most English speakers are not conscious of that difference. Hang a piece of tissue in front of your mouth while you say top and mistake; in the first word a puff of air after the T will blow the tissue out, but in the second word it will not.

The Q in qing is an aspirated CH, a phoneme we also don't have in English. Try saying cheese with that little puff of air after the CH.

This is why the old Wade-Giles transliteration system had those apostrophes and no voiced consonants. In p'ei, ch'a, ts'ai they want you to put in the puff of air, whereas in pei, cha, tsai they want you to use voiceless consonants but leave out the puff of air.

Nonetheless the American newscaster pronunciation of Beijing grates on my ears too, even though a Chinese would understand it and probably not think it's any worse than the "proper" English way, with a voiced J, because they're both wrong. :)

It's OK, I speak Chinese. :)
Another pronunciation trick in your initial learning of a new language: Study and mimic the heaviest accent in your own language, by a native speaker of the language you wish to learn. If you are a native English speaker, you can master new mouthshape combinations relatively quickly by (for example) mimicking a heavy Arabic accent in English. This helps train your mouth early to try on and get comfortable with a new pronunciation repertoire. Once you cultivate a new accent in your first native language, you can very easily graft it into a new language you're learning, and reinforce your ability to sound more "native" and fluent from the outset.
Are songs in Mandarin constrained by the tonal requirements of the speech?
The short answer: yes. That's why syllables are often spread over two notes, to convey rising and falling tones. However, in the 20th century when they started importing pop songs from other countries, particularly Japan, they translated the lyrics and just did the best they could. The vocabulary of pop tunes is pretty limited so they don't really have a lot of trouble understanding them. Soon they began writing their own pop tunes with no regard for tone correspondence. I used to go to Chinese movies with my Sichuan girlfriend in the 1970s, and even with my modest command of Mandarin I could understand some of the song lyrics. Of course it's easier for a foreigner because we never develop the same tight synapses for tonality that native speakers do. I could sometimes follow her conversations in Sichuan dialect, which has six tones in addition to some simple consonant shifts, and that's something people from Beijing have to struggle to learn to do.