No, Firenzecard does not include the Vasari Corridor, but the Firenzecard holders can purchase the “Vasari Supplement,” only at a reduced price of €20 (adults only, minors DON'T pay) and you can book it without additional costs, even if the Vasari Corridor is not included in the Firenzecard Circuit.
Reservation is also mandatory for minors.
The reservation must be made online on the website https://ticketsmuseums.comune.fi.it/1_museo-di-palazzo-vecchio/ following the steps below:
• select the date of interest and the number of participants (max 2 at a time) entitled with Firenzecard in the "Admission Free" box
• click on "check availability"
• choose and click the time of interest
• click on "Go to Cart"
• accept "I have read and accepted the Terms and Conditions"
• tick the box "I declare that I meet all the requirements for the Reduced or Free ticket(s)"
• click on "Proceed"
Yes, you need to make reservation also for all minors in the family unit associated with the adult Firenzecard holder. Reservation is free.
In case of online booking on Firenzecard webiste or through the Firenzecard app, for the Museum of Palazzo Vecchio, Stefano Bardini Museum and Novecento Museum, simply enter the number and details of the minor(s) in the appropriate fields when you make the reservation.
 
Access the Firenzecard app or go to the website www.firenzecard.
Select the desired museum, enter your Firenzecard code and the number of associated minors, choose the date and time of your visit, proceed. 
After making your booking, you will receive via email your booking confirmation to show to the museum staff at the time of your visit.

Once the reservation is made, it cannot be changed.
 
Reservation is mandatory for the following museums: Uffizi Galleries, Accademia Gallery, Palazzo Vecchio Museum, Brancacci Chapel, Tower of Palazzo Vecchio (Tower of Arnolfo).
Reservation is optional for the Stefano Bardini Museum and the Novecento Museum
You can make reservation online, through the Firenzecard app or on the website www.firenzecard.it, for Palazzo Vecchio Museum, Museo del Novecento and Stefano Bardini Museum.