How to solve these errors after running update.php?
My approach when this happens is to look at the table structure in a clean install of D6 to figure out what fields should be there, and compare that to the table structure in the current site. The issue is occurring because the update script has found something unexpected, so you need to figure out what that is. 

In some cases, I've renamed the problematic table in the current database so I've got a copy, created a new table with the correct structure, and then copied the data from the backup table into the new table. I'm on a Mac, and Sequel Pro lets you do this via a handy GUI very easily.

In others, I've looked at the specific update script noted in the error to see what changes to the database it's trying to make - comparing that to the current version of the table can help identify why it's not allowing the update to occur.

This approach usually allows me to most database issues I encounter during upgrades.