driving car in Indonesia?

No. But you can drive in Indonesia with an International Driving Permit or a local driving license.

No, it isn't valid here.

You can get a local licence known as a SIM A (for car), that being valid until your visa expires.
The new rules make it a little harder to get than it was before as you have to go to the area's head office rather than any licencing centre as before.
The good news is the corruption has disappeared.

It's worth checking locally with the authorities in Indonesia if they will accept an international driving license due to different administrative areas some don't unless it states that administrative area. I got one from the uk some time back as my local license was due to expire but wasn't recognised to use in Batam
It's been the case with many expats locally here

I understand an international licence is valid for use in Indonesia for a maximum of one year subject to approval by the police. I think that has to be a POLDA but that needs checking.

POLDA is correct. Foreigners will need to attend their local head office and the license can be issued according to their visa dates. With the correct documents and payment of  the regulated fee; the transaction is straight forward. In the case of Surabaya, you won't have any problems. I'm sure other regencies are similar and I definitely can't comment on jakarta- others are better informed. Don't forget to smile, be courteous and dress in long pants, closed shoes and collared shirt.