Sounds like your cold water cistern could be running out of water. If you can get up in the loft, take off the lid of your cistern and have a look at the water level when it does this.
You'd be lucky to fit that inside the shower casing. And if you can, you'd be very lucky to fit it without managing to kink the hose when tightening it.
All I can say is, try it. But I'm doubtful it will work and its not really ideal.
1/2"x15mm is the one you'd need.
http://m.screwfix.com/p/fischer-basin-fixing-kit/71804
A set of these will be fine for that size basin.
Is it a solid wall you are planning on fixing it to, or a stud wall?
A stud wall will require extra support obviously to have something to screw the bolts into.
No need for...
It could just be the one leak. First and easiest thing to do would be to replace the flush cone. It could just be leaking from there and dripping down onto the soil connection, making it look like two leaks. Do that first, then see if that cures it.
Is it a side entry, or a bottom entry cistern? It's not really clear to see from the angle you have taken in that photo.
Taking out the syphon and rotating it 180 degrees won't work unless there is another hole on your cistern on the left hand side for the handle.