The instructions to form 8863 cover certain circumstances when the college or university doesn't have to issue a 1098-T and says that it is possible to use those expenses for the education credits, and it lists what other documentation would be required. Expect the IRS to question the credit without a 1098-T from the second school.
cbslee is correct - no AOC for non-degree programs
As for your questions -
#1 - yes, sometimes it works like this. If the 529 is in the son's name, he reports the distribution on his return, and whoever claims the dependent with the education expenses claims the credit. In this case, it would be the lifetime learning credit.
No double dip on this - the same qualified education expenses can't be used for the credit and to offset the 529 distribution. The education credit on mom's return probably gives a bigger tax benefit that the offset to the 529, so apply the expenses toward the credit on mom's return first before using any against the 529 on the son's return. The last one I had like this, the parents income was so high that all education credit would have phased out so all of it went toward reducing the 529 on the child's return.
#2 - see my answer at the start of this post.