The SIM from the Droid likely will not work.  Is the BlackBerry Verizon branded?  If it is, and it's not currently active on another account, then call *228, option 1.  When it asks for the phone number, put in your number that you want the phone to use.

Either it will activate and you can use it with your number, or it will prompt you to call customer service to activate; they will ask for some numbers off the phone and will determine if it can be activated on your account.

As far as the contacts, you would need to have them backed up using Backup Assistant or a third party backup app and I'm not sure there even is one that will transfer contacts from Android to BlackBerry OS.  Even if they were backed up to a micro SD card (NOT the same thing as a SIM card) I doubt the formats would be compatible.