I believe so.
το παιδί - του παιδιού - των παιδιών
το σπίτι - του σπιτιού - των σπιτιών
are the two model words we learnt back in the day. (I think.) One has stress on the end, the other on the penult (I don't think there are -ι words with antepenultimate stress, for historical reasons), but in the genitive, they're stressed right on the final ending.
Final stress is also in τα παιδιά but not in τα σπίτια.
Note that, if I remember correctly, some neuter nouns in -ι do not have a genitive singular and/or plural when the stress shift would make the resulting word too "weird"... I don't know the rules for this, though, nor what they do to avoid such a missing form.
Yes, just as you both say, neuters in -ι have genetives on -ιού; -ιών but only -ί neuters have nominative pl. on -ιά.
Diminutives on -άκι do not have genitive. To say everything as diminutives is a good trick if you do not remember the gender: "Το κοριτσάκι πίνει το καφεδάκι με το ζαχαράκι", but you cannot say "the small sugar of the little girl's little coffee"???