I run all my trading thru VPS. I would be very surprised of anybody's home internet is more reliable than a professionally run VPS center. Keep in my this is there profession, keep servers running, as reliably as possible.
Unless this is also your profession, and have the resources to buy the necessary hardware, a VPS is going to be more reliable.
Yes, you can access it from anywhere.
Yes, if they want to, they can see what you are doing. There is no 100% secrecy, that's the price to pay.