Two strands of a DNA molecule are radio-labeled such that each strand can be uniquely identified. After three bouts of replication, how many new molecules of DNA are formed?

Firstly, there is an issue here with the definition of a DNA molecule. The question has defined the DNA double helix as a single molecule, where technically they are 2 molecules (with each strand being it's own molecule). I will stick with the definition implied by the question.

Each bout of replication will double the number of molecules. We start with 1, which replicates into 2 (first bout). These 2 double to 4 (second bout), and those 4 double to 8 after 3 bouts.

How many strands of DNA are radio-labeled?

This should be obvious. 2 strands are radio-labeled. It says so in the first sentence.

Are both radio-labeled strands in the same DNA molecule? Why or why not?
No. During the first bout of replication, the 2 radio-labeled strands separate and are used as 2 templates to create 2 new DNA molecules. The 2 separated strands will not come back together during subsequent bouts of replication, and will be continually be separated from newly formed DNA strands to be used as a template to make even more DNA strands.