Jen, each adult must purchase their own card: there is no single card which covers an entire family.
No, you do not make reservations if you have a Firenze card as it serves roughly the same purpose only without a pre-determined time for entry.
Your children will enter the museums with you, and any reservation fees required for them - such as the โฌ 4.00 stated for the Uffizi and Accademia - will be paid at the entrance (or to quote Rick himself: "on the spot"). The Uffizi and Accademia will have separate lines for card holders but other museums might not: no matter as those two have the long ticket queues which are to be avoided.
You skip those ticket lines with the card but not security checks: if the museum is very busy, there may be a line but it will move relatively quickly. It's also possible to them to reach the maximum number of visitors allowed inside at one time so there may be a wait for some of them to clear out.
Few others of the museums will require fees for the kids: at some you'll just collect a free ticket for them upon entry.
Bring their passports for proof of age, should you be asked.