Your sites and email shoudl be working just fine up to and during the transfer. Once the names are moved to ENom, remember to reconfigure your nameservers. This is very important if you were parking your domains with your old registrar as a freebie.
UNless the free parking WILL NEVER EXPIRE and you want to keep them there, you should update your nameservers to the new registrar. If you are hosting your site at a third party, yes it will be transparent to you . If you need to make a nameserver update, it takes some time for the changes to populate so allow 24-48 hrs. Enom used to say the wait is about 12 hrs but it's realistically longer than that. In fact it may resolve erratically during the first day or two. Don't worry.
Yes, make sure registrar lock is "off" and registrant info is IDENTICAL in loosing and receiving registrar to avoid delays.
Don't hesitate to pm me with specific questions. I do this several times a day in ENOM.
P.S. Enom also provides free parking and email redirection, so you would want to take advantage of these services and the reliable nameservers.