Hi I'm a plumber and can offer you a bit of advise.
First of, if the float (orange ball) has been snapped of the arm, then go buy yourself a new float valve assy, inc ball. Remove whats left of the original and pop into your local merchants for something that is close to indenticle ( same height, reach etc..).
Most bottom entry float valves are sized to a universal fit when looking underneath the cistern, the bit thats leaking in your case. Remember to put the washer in place before tightening the float valve assy to the base of the cistern. The washer goes inside the cistern! And take care with the amount of force you use, its only plastic! Oh and make sure the arm/float moves freely
Once you have the float valve fitted to your cistern its then a case of connecting the water pipe to it. By the looks of the 5th picture I would say the iso valve is shagged and you need a new one. ( a metal circlip would have been manufactured in place where that groove is), thats why you were able to remove the nut off the stem...not good. So buy a new one.
Turn the mains water off and fit the new iso valve and connect it to the float valve with a new washer...Job done. Turn the water back on and fill the cistern, then adjust your new float valve to give the correct water level inside the cistern, you'll see a mark impression inside the cistern.
I would strongly advise against shortening or chopping the plastic threaded part of any float valve, as you would have to do a very good job of creating a perfectly flat sealing face. If you have to wrap the thing in pufter tape to stop it leaking then you are botching it! Pufter tape is for taper/interferance fit joints. Its much easier and less time consuming to alter the length of the copper pipe to meet the plastic thread than messing with the plastic.
Have fun

Mick
Excuse my paint skills but I've slightly edited Kahns image, this is what your original iso valve connection looked like:
