If you are staying on property at a Disney resort, you can make dining reservations beginning 60 days prior to your check-in date, plus the dates of your remaining reservation (up to 10 additional days, I believe). So it gives you a jump on tough-to-get ADRs over those who don't stay at a Disney resort. So if my calculations are correct, if 4/26 is your arrival date, you can book your dining 2/26. If you arrive earlier, your booking date will be earlier.
As for Chef Mickey's, it is extremely likely you will need to break into 2 (or even 3) smaller groups for dining there. You'll probably have 2 options making them - since a person can only hold one ADR during a specific time period, you might need another member of your group to also go in and reserve a table (for 4 or perhaps 6 or 8, if available) at the same time you get it. Then the day of your meal, show up perhaps 15 minutes early and let them know you're traveling together and would like to sit together - or at least close. They will usually do all they can to get you close. With 14, it seems unlikely CM's will get you at a single table, but it doesn't hurt to ask. You can book online beginning at 6AM ET.
Option 2 for Chef Mickey's is to wait until 7AM, call 407-WDW-DINE, and have a cast member make the reservations for you - they can book more than one table for you, something you yourself cannot do.
Hope this info helps at least a little!