Can I Put Brussel Sprout Leaves On The Compost Heap?

Yes it is usually ok to put these leaves on the compost heap, so long as you tear them up and mix them with other material so that you don't just have one big 'lump' of brussel sprout leaves.
However, it is important never to add the roots of brussel sprouts to the compost heap.  This is because they will simply not rot down and you actually run the risk of spreading club root disease.  Instead they should be disposed of in a bin.  The roots of cauliflowers and cabbage also have this associated risk and they too should never be placed on the compost heap.  
The plant stems of the brussel sprouts can go to the compost heap, as long as they are chopped up very finely.  They can also be dried out and then burned.  The ash from the bonfire can then be spread around tomatoes and soft fruit to give them nutrients and 'feed' them.

Yes. You can with no problem and no worry... The compost pile will take care of all of the root disease because of the heat and bacteria that takes place... As long as you take care and turn it as it needs, there is no worry...