All of the arguments at these links go to a lack of jurisdiction to hear the challenge on the part of state ballot authorities. This would be parallel to what happened with the Great Pretender at the state ballot commission level. Similarly, all of the federal cases about the eligibility of the Great Pretender were found to be from plaintiff’s lacking standing so far as I know, with one exception. The constitutional clause does not address being put on the ballot or being nominated. Rather it speaks of “eligibility to the office.” It remains to be seen if there is a state ballot body that has the authority to address that issue or whether any person will be found to have standing should Cruz gain the office.
Here is the video of the New Jersey hearing before the appeals court.
https://www.youtube.com/watch?v=_rsbd4HVre8
The appeals court ruled that the arguments were without merit.