Hello fellow mom of twins.
You're right that you can't take your stroller with you into the lines for the rides. During our stroller years, we always had two bags with us: one giant diaper bag that lived in the stroller and one small, light backpack that we took with us at all times. The giant diaper bag would be filled with a full day's worth of diapers/wipes/bottles/clothing/snacks/toys/etc. The smaller bag would have one or two diapers, a small container of snacks, and any valuables such as wallets and cameras. After a few rides, we'd refill the small bag from the large one.
While you should never leave valuable items (tickets/money/ID/electronics/camera/phone) in a stroller parking area, you can feel quite comfortable leaving standard baby care items with your stroller during the day. I'm not naive enough to think that a diaper bag has never been stolen at WDW, but I have been there dozens of times and spoken with thousands of guests and have never heard of this happening. I myself left a diaper bag sitting in our stroller many, many times.
I'm not sure what you were asking about restaurants ... If you're asking if you can bring a stroller into the restaurant, the answer is "it depends." The restaurants generally try to discourage this. But if you have a sleeping child, and there is floor space near your table, you may be able to bring your stroller with you. You can read about an experience I had with this
HERE. As far as the diaper bag in the restaurant ... we often did bring the large diaper bag with us when dining. This helped if feeding supplies were in the big bag. But again, the choice is up to you.