How to auto deposit bitcoins on btc-e?
There is nothing in the official documentation about this sort of thing. In the examples no-one is doing anything with deposits or withdrawals either. A quick google doesn't give any results, so I think you're stuck. 

However, when I log in and want to deposit, one of the rules is the following:


  Address can be used for further deposits.


Isn't that working? Just use the same address over and over again? You will have to manually generate an address for the first time. (I know this is violating your first statement, but I used the same address at least twice for a deposit)