I think "... se po tobě stýskalo" only expresses a feeling of missing you. notice that it says "se ... stýskalo" which is third person not first. "mne" then expresses who this feeling belongs to/ who has this feeling. At least that's how I'ld interpret this sentence as a german. So maybe "To me it was really a feeling of missing you" might be a translation that keeps the idea behind "mne"
Not really. You appear to be 12 in Italian, which may work like Spanish. So consider the difference between "Lo veo." vs. "Me gusta él." While in English I am always the subject and he the direct object, the gustar-like Spanish (and perhaps Italian?) verbs switch it up, rendering him the subject and me the indirect object. (I realize the given Czech structure is not a straight up dative verb, but there is an alternative like that, Opravdu jsi mi chyběl/a. And Spanish can combine the no-fault/impersonal se with the dative, such as in "Se me olvidó pensar en ella.")
It is the case that Czech is better approached from a Romance language than from English for certain features. The dative verbs is one of them. And the positive and negative context už and ještě are a piece of cake from ya and todavía, while being a bloody mess from already, still, (any) more, and (not) yet.
As Spanish this is really useful. But I have a doubt.
In Spanish that "se" from "Se me olvidó pensar en ella" would be interpreted as a Direct Object. If I wrote it in a pure non-sense SVO estructure:
"Pensar en ella (SUB) olvidó (verb) se (DO) me (IO)" [DISCLAIMER: this makes no sense in spanish written this way xD]
So, the logic in spanish would be something like this, in English
"Thinking of her forgot itself to me" meaning "I forgot thinking of her".
So, in fact the act of "thinking of her" (pensar en ella) is the subject of the sentence. Would it mean that -if I transformed the phrase above into a pure SVO sentence-, I should write it like...
"(To) (SUBJ - "hidden") stýskalo se (DO) mne (IO) po tobe (COMP1) opravdu (COMP2)"