You can register and participate in Rajasthan 2nd round of Counselling even if you haven't participated in Round 1. Round 2 of Counselling is for those Candidates who haven't registered earlier in Round 1, Registered but did not get a seat. But you did not register, you need to register when the dates are announced. At this point in time, there is no official notification regarding Round 2 of Counselling. Please read this article for more information: https://medicine.careers360.com/articles/rajasthan-mbbs-counselling#Rajasthan-NEET-counselling-dates

No you can't because you don't fill the 1st round's application. 1st round is used for analysis of your Brain and skills, so if you don't fill 1st round's application you don't have any idea what I do, so that is necessary to fill 1st round's application.