Electrical gremlins

Hi. My car has been off the road for a few months. The battery went flat as the alarm kept going off. I have bought. A new battery but on connecting it the hazard lights, head lights, main & dip, and tail lights all come on. Also the washer jets emptied the content of the screen wash until it was empty. I tried locking and unlocking the car to see if it would sort itself out. The alarm chirped for both so I suspect that works.
I put the ignition key in and turned it to position two. This was to see if the ignition lights came on.

They did but also the starter motor engaged and the engine fired. I had not turned the key to position 3 to crank the engine. With the vehicle running it drives, goes in and out of gear but with all the lights still on as first described.

I have removed the fuse for the washer jets so that goes off but have no idea why everything else is doing its own thing.

The only way to stop it all is by disconnecting the battery!

Any ideas would be appreciated.
Thanks.

Neil