Are there rules about the differences between pinyin spelling and pronunciation?
The pronunciations of finals do not change when used after different finals, with perhaps only one exception: 'i'. It has three variations: 'zi ci si', 'zhi chi shi ri', and all others.

NOTE: Not many Chinese know the differences, but you can compare:

English pinyin
 Lee     li
 she     shi    (the two consonants are also different)
 see     si


The three english /ee/ in the above word are the same, but the three Chinese i are all different.

However, the spellings of the finals do change when used after different initials. The details can be found in 汉语拼音方案 (the Scheme for the Chinese Phonetic Alphabet), which is the official standard for Chinese pinyin.

For the example mentioned by the OP, 'ü' will be written as 'u' after 'j' 'q' and 'x'. ('u' will never be used after 'j' 'q' or 'x')

NOTE2: I don't know whether the full detail of the scheme is appropriate for beginners. There are two links in Chinese: 汉典 BaiduBaike