Advice Required!

Hey guys,

Just bought my first corsa d, a 2008 1.2 Diesel Car derived van. I bought the car cheap as it seemed to be having central locking/immobilizer issues.

The lock button on the key seemed very sensitive and would lock the car with the feintest of touches and to unlock i'd have to press it numerous times and hover the key at the windshield.

Sometime the car wouldn't start, but the issue seemed immobilizer related as the engine wouldn't even try to turn over and the fixed seemed to be to lock and unlock a few times. When it did turn over it would start first time and the car ran without issue.

After a day I started to get either the batter light or the yellow car & spanner light come on while driving and stay on (sometimes both) but periodically they would go off. The car was running with no issues however.
Then, in addition, I started getting 'F' appear on the dash above the odometer when trying to start. If that was showing it would almost defiantly not turn over.

Now, to add to these issues, the car cuts out when running after a minute or so.
When pressing the car unlock button, the car rarely unlocks and it seems to cycle through lights. One press it puts the headlights on, then its flashes the hazards, then the lights go off and repeat until eventually it does open the doors. The door lock button on the center console also doesn't do anything, but does light up when locked.

Has anyone any experience with a similar problem?

Thanks in advance!