In a sentence, if I want to say "the items of my family and I ", can I say it in this way: my family and I's items or is the correct replacement "mine". Can an explanation to why "I's" is correct or incorrect be provided?
Idiomatic would be
Those are his and his family's items.
Those are my and my family's items.
Sometimes you will hear (non-standard)
me and my family's items
We can also say
The items which belong to me and my family...
This would also be grammatical but very rare
... the items of my family and me
We'd usually say simply
... our items
There would have to be some unusual context to require our to be broken out into of my family and me.
The possessive of the first person is not formed (and has never been formed) by affixing the -s ending. The possessive form is my.
P.S. Until someone shows me evidence to the contrary, I will continue to regard phrases like "my wife and I's collaboration dinner" (and similar) as the perverse result of prescriptive approaches to teaching grammar in middle school and high school. Throughout the 20th century, schools hammered this rule into the heads of students: "When speaking of yourself and another person, always say "she and I" or "he and I" or "they and I" never "me and her". And what false idea stuck? and I is always the _proper_ second element of such a compound constituent. The speakers are trying to speak properly not naturally, and they have only a fuzzy, distorted memory of what they learned in school.