Its the pick in the amateur draft, not the rule 5. 

You will lose your first pick in the draft by signing a type A, if your first pick is within the top 15.  If not, you will keep that pick and the other team will get a supplemental pick.  If your first round pick is 16th or later, you lose that pick and won't get a first round choice.

Not sure exactly how the ordering works with signing two Type As. 

amature draft.... what if my next pick after 16 is a comp pick.... if I sign a type A would I lose that one or my natural 2nd round pick?

I'm pretty certain it is your first natural pick after pick 16 and would not take your compensation pick from you.

1)  You never lose comp picks to compensate for signing a type A; it would be your natural second rounder.
2)  The worse team the previous year, ie, the team with the best natural draft position, also gets the best comp pick.  Not sure if MLB does it that way but HBD does.

This doesn't impact HBD, but when a team signs multiply Type A picks in MLB, the team losing the highest overall rated player, as determined by Elias, gets the top compensation pick, whether that is in the first or second round; both (or all) teams get a supplemental pick.  So, for instance, when the Yankees signed Sabathia and Teixeira in the same offseason, the Angels got the first-round pick because Teixeira had a higher Elias ranking than Sabathia.