The Hebrews did not count days from midnight to midnight and different rabbinical schools had different rules for when exactly a day officially ended.*
However, I believe they used the same word, yom, when they were referring to a normal day as was used in Genesis.
Correct.
The "yom" of Genesis is considered to be the same unit of time as the "yom" used for counting the "omer" of Leviticus 23:15.