This car is still quite young - far too young I would have thought to be fit to break. My Skoda is a year older, with 10k more miles on it and it's running absolutely grand.
If you got all the problems solved with your Peugeot, would you intend to keep it? It could be that if you paid to fix it, then it could be cheaper and less hassle than buying a new car.
Gearbox trouble is either really simple or horrendous. It might be suggested that a new box is fitted and this is major work. Getting it started should be easy and any decent backstreet mechanic should be able to deal with that. I would certainly get it running because a car that won't start is pretty useless and I'd not suspect anyone will be much interested in taking on your problem unless you virtually give it away.
It sounds like the car hasn't had a decent service record - which has resulted in a few problems cropping up. This may also influence whether or not you would want to hold onto it: i.e. what else is likely to go wrong.
If anyone can suggest a decent workshop (not a main dealer as they'll charge an arm and a leg), that has experience of Peugeot, then they might be able to fix the immediate problems. You could ask them to be frank about the car's condition to give you an impression about the condition of other bits and pieces. You can decide then to sell it or hold onto it but I think scrapping it would be extreme at this stage.