so when there's an index on a column, and you do a simple SELECT * FROM table WHERE indexed_column = value, is that a O(1) search? does it matter whether the contents indexed are integers or string?
None of the lookups in MySQL's MyISAM or InnoDB storage engines are O(1) searches. Those storage engines use B+Trees to implement indexes. The best they can do is O(log2n) searches.
MEMORY storage engine uses a HASH index type by default, as well as the B+Tree index type. Only the HASH index can achieve O(1) lookups.
The data type of the indexed column doesn't change this in either case.
For more on MySQL indexes, read http://dev.mysql.com/doc/refman/5.1/en/mysql-indexes.html