Are these joists to code?
The "mini-joists", as you call them, are placed there to mitigate the overhang that is longer than 6". So if they are properly installed you would have nothing to worry about. Proper installation means that they are securely fastened to the ledger board and the cross header. 

When you have the deck surface removed in that area you may want to look closely at how the mini-joists are secured in place. It there is any concern you may want to add an angle corner bracket at the ledger end and a support hanger at the header end. (Since the spacing is extremely cramped it may be necessary to temporarily remove the mini-joist and attach the bracket/hanger to the member before putting back in place).