If you want a live video feed, I'd say definitely go with the Raspberry Pi. But it's not a matter of "well there's three 'ands' so I need the RPi". Even if you only wanted to do the live video feed, nothing else, the Raspberry Pi would still be the way to go. It's a matter of resources. Arduino boards have much less processing resources than an RPi. When you need to to something processing-intensive like video, the Arduino simply doesn't have what it takes. But the Arduino is very good at doing more simple things, even if you need it to do many simple things.
Live video feed: get a cheap security cam with WiFi built in.
Web control of the servo etc: NodeMCU or WeMOS mini. WiFi built in, can serve simple web pages, good enough for this (why do you need JS and other complexities when a bit of good old HTML can do the job?). Multiple users can connect at the same time to one such device (I don't know the limit). But why would you even need multiple users to send a command to pop one balloon?
Just make sure your servo is happy with 3.3V signals, otherwise you may need a level shifter.