I'm in the same boat as you, I'm nearing one year on my Discover IT secured card. I now have an AMEX Green, Cap One Quicksilver One, and my Cap One Platinum Secured, went unsecured with a credit increase. All 3 FICO's between 620-650.
Adding funds is also completely necessary? I wasn't aware of that until recently.
The typical credit line I see after people graduate is $2-2500 regardless of how much is secured. Some people swear that depositing more sped up graduation (which I doubt) and got them a higher credit line on graduation.
As for needing permission to get an increase, when you applied, Discover treated it like any other app and they have a cap of what they're comfortable lending you. So, if that cap was $1500, had you applied for an unsecured card, you would have gotten a $1500 SL, or on your secured, even though the max deposit is $2500, they would have only let you deposit $1500. When you want to add money, they reevaluate that cap to see if it has gone up or down. You'll be able to add money up to that cap, or the max of $2500.
And to answer your question, no it won't affect graduation date as they base it on your reports when they evaluate every month.
I started my secured card with 300.00 and added another 200.00. My 8th statement cut on March 10th no CLI not unsecured. I called in yesterday to inquire and was told I have used too much available credit. I always paid off and never late on any cards. My TU score 685 I believe it was 670ish when I applied. I paid my card to 0.00 before the due date. I informed the rep when I called in that I would close if it does not unsecure in July at the year mark. I woke up this morning to find out my card was unsecured with 250.00 increase 
whatever!!! Just give me my $500.00 so that I may bank it. I truly believe if you put 200.00 in and not increase it you do better than a good score and large deposit. Oh well I will attempt a CLI at the year mark.