I always fake both sheet and halliard and never have problems with twists or kinks, but then I built deck boxes specifically to hold them out of the way. 

I'm afraid that the twisting halliard has everything to do with the swivel blocks and hardly anything to do with the rope, although three strand would be worse than braided.  Do yourself a favour and fix those blocks.  At worst, the twist can get so bad that you can neither raise nor lower the sail.  New rope always wants to twist anyway, at first, so if you are reeving brand new rope through those swivel blocks, you might end up with a headache.

On my three last junks I have not had problems with twist in the halyard. My guess is that this has two reasons:

• There are two blocks at the masthead plus two blocks or points at the slingpoint on the yard, sitting about 20cm apart. Together these will prevent the halyard from making a big twist.
• The tail of the halyard has not been coiled in my last three boats (Johanna, Frøken Sørensen and Ingeborg). They are all being stuffed in a canvas bag, mountain-climber style.

In case a rope has developed twist somehow, I toss one end over board and tow it for a while behind the boat, before hauling it back in. 
Even the sheet is never being coiled. During sailing it is dropped randomly on the cockpit floor. After the trip, the line is stuffed in a bag, just as the halyard.

This works well for me.

Arne

One trick I have learned is if you coil a rope clockwise every time, then you introduce twist in the rope. If you alternate between coiling clockwise then coiling anticlockwise, the twist cancels out and you get a nice relaxed rope that runs easily and does not twist when hoisted.

Annie wrote:

The usual cause of halyard twist - once new rope has settled down - is allowing the block(s) to swivel.  Yours are locked, aren't they?

Nope. All the blocks are on swivels. I have fixed blocks, and I planned to use them. More recently I ordered additional blocks, this time with swivels, so I could make my mast head as much like the photos from Arne as possible. Maybe I will go back to fixed blocks next year if this doesn't work out.

I think I had problems with twist for two reasons last year.

1. I bought the cheapest double braided polyester I could find.
2. I was in the habit of coiling the rope around my hand and elbow at the end of every sail before storing it on a winch.

The old halyard is now very stiff and kinks easily. I imagine the core and the outer-braid are very badly twisted around.

edit: And only now do I notice that Arne seized the swivel blocks. darn.

Thank you for the suggestions Annie and David. For now I decided to launch the boat so I can at least try to get sailing before the summer is over. I think I will sand out the existing checks and put some thickened epoxy and paint on them sometime soon. In the coming winter I will (somehow) remove the paint and any bad wood before putting glass and epoxy over everything.

Yesterday with the help of two friends I got the boat from my house on the trailer to the mooring ball in 3.5 hours. That was a lot after already working all day.

Here are the changes so far since last season:

1. A 3 part halyard to replace the 2 part halyard I tried originally.
2. White paint on the tabernacle. (I think it looks a little better now)
3. Swim/Boarding ladder on the stern
4. Rudder cracks ground out, filled and sanded, and the complete rudder painted
5. New mast hinge arrangement
6. Replaced the halyard with this rope. I hope this will eliminate the issues with rope twist.