In Oracle how to save a sequence.nextval in a variable to be reused in multiple inserts

I'm pretty certain this will end up being an artifact of your code, or the .net driver you are using. I've knocked up a quick demo for you using pure SQL - PL/SQL and never get a lost sequence value. Incidentally the ref cursor you are using is probably unnecessary and likely impacts performance and readability of the code - my demo includes an insert_record2 procedure that consistently performs over 10% faster -in about 26s on my laptop vs 36 for the ref cursor version. I at least also think is easier to understand. You could obviously run a modified version against your test database complete with audit trigger.

/* 
demo for dbse 
assumes a user with create table, create sequence, create procedure pivs and quota. 

*/

drop table dbse13142 purge;

create table dbse13142(
    the_id number not null
,   name   varchar2(20)
,   userid number)
;

drop sequence x_seq;
CREATE SEQUENCE  X_SEQ NOCACHE  NOORDER  NOCYCLE ;

create or replace PROCEDURE Insert_Record
                (p_name    IN  VARCHAR2,                
                 p_userid  IN  INTEGER,
                 cur_out   OUT sys_refcursor)
    IS
        v_id NUMBER := 0;
    BEGIN
        -- Get id value from sequence
        SELECT x_seq.nextval
          INTO v_id
          FROM dual;

        -- Line below is X_PKG line 40
        INSERT INTO dbse13142
            (the_id,            
             name,                        
             userid)
          VALUES
            (v_id,
             p_name,                        
             p_userid);

        -- Return new id
        OPEN cur_out FOR
            SELECT v_id the_id
              FROM dual;
    END;
/


create or replace PROCEDURE Insert_Record2
                (p_name    IN  VARCHAR2,                
                 p_userid  IN  INTEGER,
                 p_theid   OUT dbse13142.the_id%type)
    IS
    BEGIN
        -- Get id value from sequence
        SELECT x_seq.nextval
          INTO p_theid
          FROM dual;

        -- Line below is X_PKG line 40
        INSERT INTO dbse13142
            (the_id,            
             name,                        
             userid)
          VALUES
            (p_theid,
             p_name,                        
             p_userid);
    END;
/

set timing on

declare
   c sys_refcursor;
begin   
for i in 1..100000 loop
   insert_record('User '||i,i,c);
   close c;
end loop;
commit;
end;
/

select count(*) from dbse13142;
truncate table dbse13142;

declare
  x number;
begin   
for i in 1..100000 loop
   insert_record2('User '||i,i,x);
end loop;
commit;
end;
/

select count(*) from dbse13142;
truncate table dbse13142;

It depends on why you are creating the temporary tables in MySQL.

Frequently, people that are creating temporary tables in other databases are doing so in order to work around limitations that don't exist in Oracle where readers don't block writers and writers don't block readers. In other databases, you commonly copy data from a permanent table to a temporary table so that your process doesn't block some other process that needs the same data. Since Oracle provides multi-version read consistency, however, this isn't necessary (or beneficial) in Oracle-- your process can process the data sitting in the real tables without worrying that it is going to block someone else. If that's the situation you're in, the proper response is simply to remove the temporary tables and process the data from the permanent tables.

Assuming that you really need a temporary copy of the data, you can create global temporary tables. You would create these tables outside of your code just like a permanent table and use them just like a permanent table inside your code. The global temporary table ensures that each session can only see the data that session has inserted. The only difference is that you're not dropping and recreating the structure of the table inside your procedure.

Another alternative would be to pull the data into a PL/SQL collection that you work with rather than using a temporary table. PL/SQL collections are stored in the server's PGA (one of Oracle's memory structures) so you generally want to limit the size of the collection particularly if there is a lot of data to process or there could be many sessions processing data simultaneously. You can do something like

DECLARE
  TYPE emp_tbl IS TABLE OF emp%rowtype;
  l_emps emp_tbl;

  CURSOR emp_cur 
      IS SELECT *
           FROM emp;
BEGIN
  OPEN emp_cur;
  LOOP
    -- Fetch 10 rows at a time from the cursor into the collection.
    -- You'd realistically want a larger limit, something between 100 and 1000 generally
    FETCH emp_cur 
     BULK COLLECT INTO l_emps
    LIMIT 10;

    EXIT WHEN l_emps.COUNT = 0;

    -- An example of manipulating the collection in memory
    FOR i IN 1 .. l_emps.COUNT
    LOOP
      l_emps(i).sal := l_emps(i).sal * 2;
    END LOOP;

    -- And an example of using the collection to update a table
    FORALL i IN 1 .. l_emps.COUNT
      UPDATE emp
         SET sal = l_emps(i).sal
       WHERE empno = l_emps(i).empno;
  END LOOP;
END;

Of course, in the example above, it would be far simpler (and more efficient) to simply issue an UPDATE statement that would double every employee's salary.