I don't understand how all of your credit card transactions are showing up twice. The only thing that shows up in multiple places in my mint account are transfers to my Vanguard account, but I just ignore them. You could classify the debits to your bank accounts as "Savings" though, so that you'll see that label when you look at your spending. Where do you see the transactions twice? Have you checked your accounts to make sure that you don't have an accidental duplicate? If you do, you can delete it or just make it inactive. I had that problem when I first signed up.
I also find that the "Transactions" tab is basically worthless to me. I stopped paying any attention to it almost immediately. I only ever view the "Trends" tab using the pie chart view, where I can look at the overall spending and then click on specific categories to see specific details and transactions. When I pay my credit card, the payment shows up in "Transactions" but doesn't show up in the "Trends" categories (because that money has already been accounted for).
IMHO, service fees ARE spending, and Mint is correct to classify them as such. Mint treats dividends as income, though, so once again, they show up under the "Transactions" tab but won't show up in the "Trends" for spending -- only in the "Trends" for income.