I know that ございます is used in informal settings. Still, I have some confusion as we use it after "Good Morning" and not "Good evening". Are there any specific rules related to it's usage in Japanese language? Also, one more doubt is when to use "Gozaimashita" in place of it.
These are set phrases and should probably just be learned that way:
You see ございます in the first two because they are formed off of adjectives of a particular form.
There are entire rules on how to derive these forms. For example,
うつくしい -> うつくしゅうございます
But also it seems that how these rules are being used has shifted over the past several decades. Learn these as set forms to use in the circumstances called for. The only time I recall hearing うつくしゅうございます was in tea ceremonies offered by a middle-aged woman who was always super-super polite.
The difference between ございます and ございました is just a matter of non-past and past tense. This is a whole topic unto itself. Also as to when to use ありがとうございます vs ありがとうございました is an entire separate topic too.
The standard copula is だ which can also be expressed as です as a politer form.
である is an older form of だ. You'll see it in literature and in speeches and such. You'll sound strange using である unless the circumstances are rather specific.
でございます is a politer version of です but derived from the である form. Essentially, ござる is the humble version of ある. ござる becomes ございます.
I'd say to refrain from using ござる itself unless you want to sound like you're in a samurai movie. And, as for ございます, if you're a beginner, just use it as you hear it used by others. It's really that simple.
Unless you're in Japan studying ikebana, tea ceremony, or martial arts or in a Zen temple, the circumstances in which these forms are called for are quite limited. (At least for a beginner.)