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The baidu baike (http://baike.baidu.com/) often has the abbreviation listed under the main entry. 中华人民共和国国家发展和改革委员会, for example, has 国家发改委. 中华人民共和国国务院办公厅 lists both 国务院办公厅 and 国办. And yes, 家用电器 points to 家电 as the simpler form, although it is somewhat hidden in the entry.
Look for news articles, especially those from Xinhua, which are considered "official". Typically if an article contains a long-winding name like "人力资源和社会保障部", it is bound to introduce an abbreviation to reduce the verbosity. If you cannot find such articles, which should be unlikely for well-known entities, I suggest that you not abbreviate, especially not ...
There are many accents, but I will try to describe the pronunciation. I don't know phonetic characters, but if you go by an American accent, 去 sounds a lot like "chew" if one were to say it fast, adding more of a "ts" sound at the beginning, with a downward inflection, and emphasize the "ee" sound. Just listen to people talk, and imitate them.
The following is based upon an English accent. The u in qu is defiantly more of a goo sound, as in the sticky green stuff. As 去 is 4th tone, it is also quite a short sound. This is different to 不, where the u is more oooo, interesting. The q is more of a choo choo sound, rather than a chip sound.
I think it is a terrible mistake that the website has made, because there is no occasion when qu is pronounced tsʰu in Mandarin. Since you can actually tell the difference between u and ü, things should be easier for you now. You can just memorise that after (pinyin) j, q, x, y, ü is always written as u, and if you see u after j, q, x, y, it's always ...
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