A Merit Badge Counselor is a district membership, not a unit one. That holds even if the counselor has declared that he is only available to the Scouts from a single unit.
As with most BSA registrations, unless/until someone contacts you, and tells you there is a problem with your registration, you should consider yourself registered, in whatever position you filled out the application for, as soon as the paperwork is given to your council.
The BSA membership cards are usually only printed out once a year. That is usually right before recharter time. The only time you will need your ID number is for online training. You can contact your council Registrar in about a month, and ask for your ID number. You should, possibly, hopefully, be in the wonderful BSA database that is ScoutNet by then!
Once you get your ID #, go back into the BSA training site and update your information with your ID.
The Scoutmaster provides Blue Cards to the Scouts in his/her Troop. The Merit Badge Counselor signs the cards signifying completion of requirements. The Scout keeps track of his cards.
You should check with your council to see when their next Merit Badge Counselor training is being held. Meanwhile, these links might help you -http://www.scouting.org/sitecore/content/Home/BoyScouts/GuideforMeritBadgeCounselors/MBCounselorGuide.aspxhttp://www.scouting.org/sitecore/content/Home/BoyScouts/GuideforMeritBadgeCounselors/IntroToMeritBadges.aspx