Dear customer reading your question, I understand that it is a problem of transmission, the problem is that when we buy a second hand vehicle, we do not know exactly the maintenance history, I see that it is 2010 and it is very likely that it was already due to do the maintenance of the transmission, like replacement of filters and oil but many times when someone wants to sell the vehicle does not do this type of maintenance as it always is an extra cost to perform them.
About the type of failure you present, it is likely that your Peugeot has problems in the oil viscosity because they have not been regularly replaced.
The viscosity of the oil can cause the transmission not to make changes or remain neutral because these transmissions work with oil pressure differences to allow displacements
So basically you should start at this point by maintaining the transmission that the old owners probably did not do.
On the other hand, Your peugeot works in the transmission with a set of selenoids and electronic valves that are responsible for activating or deactivating gears. What happens is that also the lack of maintenance can cause the selenoids and the electronic valves to remain stuck. The problem in neutral points us thos type of problems But it may be a major failure that can merit a disarmament of the transmission for repair which I hope is not your case because these types of repairs are always somewhat expensive
This is to give you an idea of the most common factors that cause you this type of failure but it can also be a problem at the level of the electronic module. It may be a type of failure at the transmission configuration level as an error code stored in the module which must be verified through a diagnostic device
Then, this type of failure may be due several factors but i do not consider it a problem in driving how they are suggesting you. The good thing is that your vehicle must be under warranty and all this type of work must be covered by the dealership where you bought your Peugeot
Relating your fault with the problems in the indicators, i observe a configuration problem which you will be able to solve through a diagnostic equipment as well.