To be clear: Mars does have a magnetic field (weaker and less complex than Earth's), and it does have an atmosphere (thinner than Earth's).
1.
This sounds nuts. Nuclear warheads distribute radiation. It's sad how many have been detonated on Earth. If we're going to warm up the poles, hopefully we can do it with something else that doesn't produce radiation.
2.
I don't think burrowing is a method of terraforming, but just a method of inhabitation. It has the advantage that there would be less UV radiation from the Sun underground.
3.
There are small animals that can survive in space (such as tardigrades), it's possible that some could survive on Mars with some atmosphere too. This seems like the most interesting option. The obvious problem is that if there is life on Mars, this might ruin our chances of finding it.
Another thing worth noting: The reason we haven't done these things yet is because we are still learning about Mars, and it would be really expensive. I could see the earliest stages of this happening in the 2020's with SpaceX or 2030's with NASA.
The Moon is pretty interesting because it has a partially molten core, it is made out of almost the exact same things that the Earth is made of, and it's really close to us compared to everything else. It's low gravity makes it easy to land and take off, but with essentially zero atmosphere, terraforming it is a huge task. It also doesn't spin - the same side is always facing the Earth - which makes for a unique set of challenges. Just some thoughts.
Europa or Titan might also be worth looking at.