The peculiarity also occurs in other Slavic languages.
Not only. You may add the Hungarian Miklós to your list.
What's its version in Serbian or Croatian?
Are there any words where "DZI" pronounced as "DZ + I" instead of "DŹ + I"?
You should understand the concept of softness in Polish first. Soft consonants require an 'i' after them that would represent their softness, if the softness isn't represented on its own by the diacritical mark ' above such a vowel. Words like 'silos', for example, are exceptions that are specially marked in dictionaries.
DZ as a hard consonant will always take 'y' instead of 'i'.