If stock A created a separate company and gave me shares of stock B, then merged with another company sending me a check for the shares, would my cost basis be the value of stock B when they issued them?
What did you pay for shares in Stock B? If you paid nothing extra, then your basis is $0.
If some of your shares in Stock A were converted to Stock B, then your basis is what you paid for Stock A.
For every stock spin off, you must allocate basis of the original shares to the new company spun off. Normally the companies have something on their websites to walk you through how to allocate basis in spin offs as it's based on the price the shares trade at the next day and often complicated.
If you post the companies involved, you could get better assistance. Do it as a new post though, don't reuse older posts.