It was only a political problem at the time because of the various treaties we had signed with Russian and the UN. The Limited Test Ban Treaty prevented it.
Yes, originally it was using nukes to launch, but was changed to using chemical rockets to orbit before it was canceled. One of the largest factors in cancellation was the Air Force which had other ideas like Apollo.
Collision is always a problem. However there were/are several schemes for overcoming this problem.
If you never try you never will.
I need to find that video I saw on traveling to Proxima Centari and link to it here. One of the options was a ship just over 1 km long. IIRC it would have 100,000 H bombs on board, and that would be sufficient to propel it to 0.1C. It would be pretty cool to watch it during the acceleration phase.