Should I change my old datetime columns from SQL Server 2005 to just date datatype for SQL Server 2012
In SQL Server 2012, date takes 3 bytes while datetime takes 8 bytes.
If your table has millions of millions of rows, then yes, then the IO and logical page size should be reduced and your performance will be quicker.

First though i'd suggest to try dumping the table into a testing table, convert the column, apply your indexes and benchmark using real life workloads against this table using the original as a baseline.