Chinese emperors had three different titles:

Era names (年號): These were two character titles (the early Song had four character titles) used by the emperors when they were alive. The era name typically symbolized the desire of the ruling emperor, for example, Shunzhi (順治) meant "Obedient Rule" and it was used because the new Qing Dynasty wanted stability and obedience from its subjects after having conquered China proper. Era names were also used for dating. For example, the year 1662 is the first year of Kangxi (康熙元年).

Posthumous names (諡號): Posthumous names were given to emperors after they died and typically ended in Huangdi (皇帝) or shortened to just Di (帝). The characters in posthumous names began increasing as time progressed. In the Western Han, emperors typically only had two characters - Emperor Wu was Xiaowu Huangdi (孝武皇帝) which was shortened to just Wu Di (武帝). By the Qing, emperors had twenty-one characters in their Posthumous names. The Shunzhi Emperor's full posthumous name was "The Emperor of Order who Observes the Heavenly Rituals with a Solemn Fate, Destined to Unify, Establishes with Extreme Talented Insights, Admires the Arts, Manifests the Might, with Great Virtue and Vast Achievement, Reaches Humanity, Purely Filial" (體天隆運定統建極英睿欽文顯武大德弘功至仁純孝章皇帝). This is usually shortened to just Emperor Zhang (章皇帝).

Temple names (廟號): Also given after death, the temple name was more exclusive than the posthumous name and always consisted of two characters. This was the name that went onto the spiritual tablets that were worshiped in the imperial ancestral halls. Temple names either ended in either Progenitor (祖) or Ancestor (宗). The Progenitor character was typically reserved for the founder of the dynasties - Li Yuan (the founder of the Tang) was Gaozu (唐高祖) and Zhu Yuanzhang (the founder of the Ming) was Ming Taizu (明太祖). Sometimes, the Progenitor title was given to the descendants of the founder who established their own lines, which would have made them equals to the founders. Khublai was Yuan Shizhu (元世祖), because it was he who officially founded the Yuan dynasty and subsequent Yuan emperors all descended from his line.[1] The Ancestor character was given to the descendants of the Progenitors. Li Shimin was Tang Taizong (唐太宗) because he was Li Yuan's son. Li Zhi was Tang Gaozong (唐高宗) because he was Li Shimin's son.[2]

[1] There are some special cases concerning the character Progenitor. Usually, people who should have it doesn't want to carry it, because they want to be seen as the legitimate successor instead of equals who established a new line. Zhao Kuangyin (Song Taizu) was succeeded by his brother instead of his son, but his brother is known as Taizong even though since he founded his own line, he should carry the character Progenitor. Zhu Di (son of Zhu Yuanzhang) did the same after he usurped the throne from his nephew. He was known as Ming Taizong until the late 1520s/early 1530s, when his temple name was changed to Chengzhu (成祖). The Qing is unique in that it had three Progenitors - Nurhaci was Qing Taizu because he founded the Later Jin (the precursor of the Qing Dynasty), Shunzhi was known as Shizhu (世祖) because he was the first Qing emperor to rule from China proper, and Kangxi was known as Shengzu (聖祖). Since Kangxi was Shunzhi's son, he should be known as an Ancestor, but the reason he was given the Progenitor title was that he consolidated Qing rule and his contribution was strong enough to warrant it.

[2] When referring to a deceased emperor, it is very common to use the temple name. Sometimes, a character from the posthumous name might also be added in. For example, the Kangxi emperor can alternatively be called Emperor Shengzu Ren (聖祖仁皇帝).

So as to your question of why titles of emperors changed, the answer is actually quite simple. It had everything to do with simplicity. Before the Tang, the emperor's posthumous names were quite short - Wen, Jing, Wu, Yuan, Xiaowen, Xiaowu, etc., so it was convenient to use them. From the Tang onwards, the characters for posthumous names became longer and longer and so it was no longer practical to use them. Consequently, temple names were adopted since they were only two characters. Era names were not used because most emperors had multiple era names during their reigns. From the Ming onwards, emperors only had one era name, so era names were used instead to describe their reign. But this is only the case in Western academia. In China, for example, temple names are more commonly used in academic writing, even for Ming and Qing emperors. So an American or British historian might describe Zhu Yuanzhang as Emperor Hongwu, but a Chinese historian would say he was Ming Taizu.

Wrong. "Di" is their posthumous title, all emperors have this title except special cases. "Zong" is just one of the more popular temple names. For example, the second emperor of Tang dynasty has temple name "太宗" but his posthumous title is "文武大圣大广孝皇帝". See the "Di" in his title and the "Zong" in his temple name? His father was "高祖" and his posthumous title is "神尧大圣大光孝皇帝". His temple name didnt have the "Zong" but has the "祖" which mean forefather. Only founder of a dynasty can have that word in their temple name, the others usually stick with "Zong" which means "the one of the previous generation".

As for some emperors usually be called with their posthumous title and some other called with their temple name, it's basically just a habit and the temple name is usually shorter than posthumous title. Some notorious emperors have posthumous title so long that nobody bothered, especially later dynasty ones. For example, Liu Bang is usually mentioned as "太祖", I think most Chinese dont even know or remember his posthumous title.

Oh of course you dont call emperor by their real name unless you want to die.