I've driven friends (who were first time visitors to Sydney) across the SHB as that (and walking it) was on their bucket for Sydney. Always gets a positive result.
Yes, The south-east tower. I usually cross the bridge from The Rocks to Kirribilli and back half a dozen times a year. Have lunch or dinner at the Kirribilli Hotel.
You can always go over the bridge for about $200. My wife has done it a couple of times but I don't have the courage.
Bridge Climb is damn expensive. I once read that the licensing fee paid by the operators of Bridge Climb (to the NSW Govt) covers the annual maintenance fee (wages of workers, paint etc) of the bridge.