Are you referring to this video?
If so, you are partly correct that "Gotta get some of that rain" is a contraction for "I have got to get some of that rain".
In this case, the speaker is showing his desire or compulsion to go out and get wet in the rain. Note that, considering his friend's appearance, this exuberence is possibly due to intoxication.
There are other uses of gotta that are similar in form, but do not necessarily evoke optimism or enthusiasm. Consider:
"Gotta love this rain, huh?"
...which is more likely to be said in a sarcastic fashion, i.e. the speaker does not like the rain.
It is possible that the singer in the video is also being sarcastic and actually means the opposite, but I am inclined to believe he is simply happy about being in the rain. In either case, it doesn't change the meaning of gotta.
In "Gotta love this rain, huh?", why do we say 'gotta'? Not 'gonna' for instance.
Gotta is a casual speech contraction of "[pronoun copula] got to...".
Gonna is a casual speech contraction of "[pronoun copula] going to...".
The use of huh? is a verbal cue for the question (essentially an invitation for the listener to agree or disagree with the preceding statement)—you wouldn't write it formally, you'd only write it to describe the pattern of speech—so the phrase "Gotta love this rain, huh?" parses as someone saying "You have got to love this rain, haven't you?":
Gotta ) ( huh?
----------------> love this rain, <-------------
You have got to ) ( haven't you?
It doesn't work to use gonna in this context, because that would parse as "You are going to love this rain, aren't you?", which is a completely different question (and one that is awkwardly phrased).
Is it because it's inevitable (talking of rain), so we have no other option but to enjoy it or being sarcastic about enjoying it?
That is certainly one reason why you might employ the phrase, "Gotta love this rain, huh?", but it does not mean you have to use gotta. You could appropriately say the expanded form, "You have got to love this rain, haven't you" or any reasonable contraction ("You've got to love this rain, huh?" and "You've gotta love this rain, huh?" are both fine); or even the literal, "I love the rain" or "I hate the rain", if you were avoiding whimsy altogether.
The use of gotta is merely a contraction employed by people with certain accents, for more comfortable speech.
In the case of the speaker in your video, a key fact is that he is singing to himself, so he is naturally going to employ a speech pattern that is more comfortable, to fit his rhythm and mood.
And for example in "You're gonna love it!" the person we are addressing still has a chance to escape what we offer or expect him\her to do?
This isn't relevant to the use of gotta (or gonna). "You're gonna love it" is just a comfortable contraction of "You are going to love it."—whether or not you are actually going to love it is a matter of storytelling, not word-use.
A person might say "You're gonna love it" in an attempt to convince a reluctant listener to join in with an activity. They may say it sarcastically, knowing that the listener is definitely not going to enjoy the activity. They may simply be wrong, having misjudged the activity or the listener. However, in all cases, they could just as legitimately say "You are going to love it", with any or none of those implications.