Hi there! I've applied for the PLSNP/NLC summer program for this upcoming summer and had similar questions to you and called the office and got their answers directly.
I don't know of anyone being denied however, I know that priority is given to those who have a conditional acceptance over those who have unconditional acceptance. I also saw on the website that priority is given to those that apply before March 1st.
For the LSAT score, I sent a screenshot of my LSAC account with the test dates and scores (and they said that was perfectly fine) but I know that you can send a copy of you LSAC email if you have it available.
For reference letters, I sent 3 just to be safe- I sent two academic and one community.
For transcripts, they told me that you can use your unofficial ones for now (they allowed me to send a copy via email) and then send the official ones directly once you've completed your coursework.
Hope this helps! Best of luck and maybe see you in the summer!