Does a UNIQUE constraint automatically create an INDEX on the field(s)?

Should I define a separate index on the email column (for searching purposes), or is the index is "automatically" added along with UNIQ_EMAIL_USER constraint?

CREATE TABLE IF NOT EXISTS `customer` (
  `id` int(11) NOT NULL AUTO_INCREMENT,
  `user_id` int(11) NOT NULL,
  `first` varchar(255) NOT NULL,
  `last` varchar(255) NOT NULL,
  `slug` varchar(255) NOT NULL,
  `email` varchar(255) NOT NULL,
  `created_at` datetime NOT NULL,
  `updated_at` datetime NOT NULL,
  PRIMARY KEY (`id`),
  UNIQUE KEY `UNIQ_SLUG` (`slug`),
  UNIQUE KEY `UNIQ_EMAIL_USER` (`email`,`user_id`),
  KEY `IDX_USER` (`user_id`)
) ENGINE=InnoDB;

EDIT: as suggested by Corbin i queried for EXPLAIN SELECT * FROM customer WHERE email = 'address' on empty table. This is the result, i don't know how to interpret it:

id select_type type possible_keys key  key_len ref  rows Extra
1  SIMPLE      ALL  NULL	      NULL NULL    NULL 1    Using where

While adding an IXD_EMAIL to the table the same query shows:

id select_type type possible_keys key       key_len ref   rows Extra
1  SIMPLE      ref  IDX_EMAIL     IDX_EMAIL 257     const 1    Using where

A unique key is a special case of index, acting like a regular index with added checking for uniqueness. Using SHOW INDEXES FROM customer you can see your unique keys are in fact B-tree type indexes.

A composite index on (email, user_id) is enough, you don't need a separate index on email only - MySQL can use leftmost parts of a composite index. There may be some border cases where the size of an index can slow down your queries, but you should not worry about them until you actually run into them.

As for testing index usage you should first fill your table with some data to make optimizer think it's actually worth to use that index.