Starter/ Wire/ Electrical problem with my Peugeot 307

I am a new driver and this is my first car.... I am quite clueless about cars, will appreciate any advice please...mine is a 307, 2007, petrol, automatic, 1600cc.

The problems of my car:

Main problem:

In the past month (since I bought the car from a trader in Reading), twice I couldn’t start the car and needed to start the engine by jumping the battery. Initially I thought it was the problem of the battery but the guy from Halford told me that my battery seems fine. I went to check the starter, any my starter seem fine too and the garage said it could be a electrical problem which they couldn't really confirm as they don't have the machine for that.

I have other (minor) related issues with the car:

1. Occasionally, when I start the engine, some or all the error lights will go up, sometime the engine’s one too… If this happens, my headlight and screen wiper will be switched on automatically and needed to be switched on and off to switch them off.
   
   
2. I have replaced the battery in my key remote and try to sync it to my car using ways suggested online, but it still doesn’t work… not sure if there is anything I could do about it.
   
   
3. Central lock: to unlock the all doors of my car, I have to put the key in and turn to the non-engine-starting position, is that normal? With the key itself, without plugging it into the ignition place, I could only open the driver door (I thought all doors will be unlocked, but I am not sure if my car should do that as it is a fairly old model).
   
   
4. Radio’s time and date always reset itself after each drive
   
   I will be ultra grateful for any advice... hope can resolve my problems without costing lotssss