I'll just add where my answers are different:
2. Do you accelerate transaction you have had no control of mining fee?
This is the only reason I've used it for myself. I do maintain one or two online wallets. Previously, I was moving away from them but with the way the mempool looks, as well as high fees set to be here for a while, I've looked to keep at least one wallet online. It removes the headaches of receiving only daily inputs. Unfortunately, no control over fees can mean that sometimes withdrawals include fees too low. I've turned to accelerators in the past, with success.
3. Do you use free accelerator to save money on mining fee?
No, when they were easily available and working, fees weren't that high anyway. With the way they're all queued up though, you still have to pay a reasonable amount to have a chance of prioritised (talking about antpool).
4. Do you have a control of input private key or output private key or none?
My answer to #2 means I don't. Eventually, everyone will have to use their own wallets where they control private keys... but then we're in trouble if we often have to receive $10 to $50 payments. Looking forward to off-chain settlements!