It's similar with Italian: mi piace/piaciono means 'I like', depending on the number of the thing that is liked. In Italian and Greek, the subject is the thing that is liked while the person who likes it is expressed in the dative case. So in Greek, you can say μου αρέσουν τα γάντια, which means 'I like the gloves', but if you want to say 'I like the glove' it's μου αρέσει το γάντι. Literally you're saying 'the glove pleases me' or 'the gloves please me'.
In an important site "Greek as a foreign language" of University of Athens there is an extensive explanation of the behavior of μου αρέσει. http://www2.media.uoa.gr/language/grammar/details.php?id=75
1) Verbs of this type have the following peculiarities: their subject is semantically more like the object, while the semantic subject appears as a thin type of personal pronoun in general. The subject agrees with the verb in number:
Μου αρέσει το σχολείο
Μου αρέσουν τα μαθήματα
You can not display anything between the pronoun and the verb:
*Μου το αρέσει
*Μου το πράσινο αρέσει
*Μου αυτό αρέσει
It is possible to display the whole noun phrase (or strong type of pronoun), also in general, which have the same reference to the weak form of the personal pronoun. Even in this case, the weak type must appear in the same place:
Του Γιάννη του αρέσει η αριθμητική
Εμένα μου αρέσει η γεωγραφία
Του αρέσει η αριθμητική του Γιάννη
Μου αρέσει η γεωγραφία εμένα.
NB .: attention to changes of intonation in each case
And more, more advanced, as the use with the verb.
Yes, του is the short form of the pronoun αυτός (genitive: αυτού or αυτουνού)
Μου αρέσει is an irregular verb and it gets genitive case. I don't know why, but I can guess it is due to an old version of the Greek language. I will search it and I will tell you if I find something related to it :)
You are right - in ancient greek αρέσει was αρέσκει and and the person pleased went in the dative case, producing a sense like the English 'it is pleasing to me" . By about the year 1000 Greek had dropped the dative case and was using the genitive as a replacement, as it does today. Italian still uses the dative - see Hermesianax above.
Short answer: both.
Long answer: it's a loanword and I'm afraid it's one of those words whose correct spelling (μπίρα), decided one day by academics, didn't really catch on. Maybe it's taught in younger generations? You can also check google trends to gauge how their frequencies compare.