Is it possible to make a follower a member of the blades, yet still be able to have them follow you to aid you on your quests?
Yes you can still use the follower after you've admitted them into the Blades. Simply find them at the temple and speak to them as you would any other follower to bring up the standard options for getting a follower to join you.
Followers you've recruited into the Blades will generally remain at the temple when they're not accompanying you, with few exceptions.
Do note that when you admit a follower into the Blades you lose them as a follower while they are doing the admittance ceremony. However, immediately following that, they should be available to re-recruit.