I have interpreted your question ‘What are the stories in 1001 nights’ to mean ‘What are the specific titles of the numerous stories contained within the work’? Now since it would be time consuming and tedious (though not impossible) to dive in and list each story myself, I will not be doing that and instead will simply point you to a site where all the hard work has been carried out already.
There is a Wikipedia page detailing each of the stories appearing in Richard Burton's The Book of the Thousand Nights and a Night here and another webpage which is more comprehensive, not confined solely to Burton.
Secondly you ask for a “modern version” faithful to the original Arabic. Two choices are: 1. Yasmine Seale’s The Annotated Arabian Nights, published by Liveright, November 2021, and 2. The Arabian Nights by Muhsin Mahdi as translated by Hussain Haddawy – publ. Knopf-Doubleday 1992.
The most convenient translations, on account of being both free (no cost) and accessible online, are those by Richard Francis Burton and John Payne, both available at the Internet Archive, Project Gutenberg and Wikisource.
It is to be noted that Burton and Payne in addition to using Arabic sources both borrow from Antoine Galland's French translation published between 1704 and 1717, which was the first in a European language and contained additional stories passed on to him orally by a Maronite story teller from Aleppo (Hanna Diyab), which are not found in documentary sources.
The Arabic version that has the greatest claim to being original is a 14th century manuscript at the Bibliothèque national de France. Muhsin Mahdi based his 1984 version of The Arabian Nights upon this oldest surviving Arabic manuscript, referred to as the 'Galland manuscript'.
(I acknowledge the significant time interval between my answer and the two previous answers, and I suspect in the space of almost five years new developments beyond the 2018 context would have taken place which may account for the variation).