Indexing every column in a table

1) Yes, of course.
2) No, it doesn't defeat the purpose of index. Just remember that mysql can't use more than 1 index per table and that adding more indexes slowdowns insert/update/delete operations. So avoid creating indexes that aren't used, create multicolumn indexes that matches your queries best instead.

Indexing any table, either memory or file system based, will speed up queries that select or sort results based on that column. This is because the index works like a tree structure and the search distance depends on the depth of the tree, which increases a lot slower than the row count of the column (logarithmic).

Indexing every column does not defeat the purpose of the index, but it will slow up inserts and updates because those changes will cause an update of every index of that table. Also, the indexes take up space on the database server, so that is another drawback to be considered.

Other SO questions to read relating to this question:

Best practices for indexing
What is an index
How many indexes are enough