If you have a cable broadband connection (eg Virgin) you can keep your existing number. If you've got ADSL broadband (eg BT, Sky, TalkTalk), the process can be more complicated but in the majority of cases, your existing number can be transferred. You can find out more here
The process can take between two weeks and a month - it is important that all the required information is provided speedily and accurately to avoid any delays. In the interim, you will effectively have two phone numbers to use, your original landline number and your new, temporary Vonage number.
No. Your original phone service needs to be active for the number transfer to take place, and if it is cancelled, you will lose your original number. When your number is transferred, your original service will automatically be cancelled.