Would it be correct to say stuff like:
Mir hat angst -> I am afraid / feeling fear
Mir ist gluecklich -> I am afraid/ feeling happy
In analogue to the idea of "Mir ist Kalt"?
Would it be correct to say stuff like:
Mir hat angst -> I am afraid / feeling fear
Mir ist gluecklich -> I am afraid/ feeling happy
In analogue to the idea of "Mir ist Kalt"?
It would be wrong.
The construction is limited to certain adjectives. Your question seems to imply you assume that there would be a certain kind of "logic" here. There is none. Your constructs could theoretically exist. But in the German that happens to be spoken, these constructions just don't exist.
Also notice that your suggested pattern has hat with and adjective, but the pattern in German is to use ist with an adjective.
There is no inherent necessity to all of this. It is a historical, a contingent fact. This contingency is also illustrated by the fact that other languages have similar expressions following a different pattern:
Languages follow some rules sometimes, but it is arbitrary when they do. You cannot learn a language by making up some rules and hoping that they are going to represent the language that is actually spoken.
I am not sure I can give the comprehensive list. The following examples come to my mind:
Mir ist kalt / warm / heiß.
Mir ist langweilig.
Mir ist schlecht / übel.
Even if this list might not be exhaustive the examples from your question are definitely not on the list. The chance to hit one of the idiomatic expressions is quite small when using your analogy as a recipe for creating expressions.
A more productive construction is:
Mir ist [...] zumute.
meaning "I feel [...]", "I am in an [...] mood."
Mir ist ängstlich zumute.
Mir ist traurig zumute.
Mir ist blümerant zumute.
Mir ist komisch zumute.
Mir ist zum Weinen zumute.
But also with this construction there is a limited list of phrases that are idiomatic. I am not able to exhaust the list. :)