Regarding your first question, there is no difference of the svayamvara ceremonies of Draupadi and Laksmana, they were almost alike. The only difference was that the fish was covered with a cloth in the case of Laksmana, but the fish was naked in the case of Draupadi.
Draupadi (Letters)
Correspondence
1970 Correspondence
1976 Correspondence
Letter to Gadi -- Mayapur 16 February, 1976:
Regarding your questions; Arjuna, Yudhisthira Maharaja, and Draupadi were not killed in the battle. They are always personal associates of Krishna. Wherever Krishna goes, they go. The Bhagavad-gita says: bahuni me vyatitani janmani tava carjuna. "Both you and I have passed many lives; I remember, but you do not."