Yes the Mangal dosha get cancelled there is nothing to worry on that front. Regarding checking for Nadi and Bhakoot, it is really a long process and it takes lot of time to study divisional charts. I suggest that you may come through my paid analysis for the same which will give you a detailed horoscope matching analysis. Remedies would depend upon if the above doshas are getting cancelled or not, so it is premature to reply now.
Kindly note that it does not matter if you follow traditions and get married as per your customs or you do a court marriage. When we are looking at the Horoscope matching we are looking at detailed analysis man and women relationship which covers mental compatibility and also physical relationship. The marriage compatibility even looks at temperament issues, Love relationship, sexual relationship, Children, financial , luck and destiny. All these are required when you married, they are not needed when you are in a simple relationship or as we can say dating.