And if he was a Kohain, she should not live in the same street with him - ואם היה כהן לא תדור עמו במבוי
A ישראל may live in the same שכונה as his former unmarried wife. If she remarried they may live in the same מבוי but not in the same שכונה, this applies to an אשת כהן as well. However by a כהן if she is still single he may not live even in the same מבוי as his former unmarried wife.
These are believed to testify as adults what they saw in their childhood - אלו נאמנין להעיד בגודלן מה שראו בקוטנן
בדיקת חמץ is somewhat בידו, therefore נשים עבדים וקטנים are believed only because it is a דרבנן; by a complete בידו like ניקור וחלה, a woman is believed even by a דאורייתא; where it is not בידו at all, only גדולים can testify what they saw בקוטנן (and only by a דרבנן).