Difficulty does not support price - that is right - this happens because supply of bitcoin does not depend on difficulty (in long term). But when that supply is halved - then we can expect the price to go up.
Exactly. Bitcoin currently mints 7200 coins per day. At $1 USD that means a $7200 USD net inflow needs to come into the economy to avoid prices falling. Technically it is less than that because some % is hoarded but lets ignore than for a second..
Bitcoins are like any other commodity. Right now we are at $2.40 USD:BTC. If daily demand for Bitcoin is >$17,280 then more dollars are chasing the same number of coins and prices rise. If daily demand is <$17,280 then less dollars are chasing the same number of coins and prices fall.
Note this is on a long term fundamental basis excluding manipulation. So when the reward drops in half @ $2.40 it won't require a net inflow of $17K daily it will only require a ~$8K daily net inflow.
On price:The reward cut may not result in prices rising but it will put upward pressure on prices.
If prices rise or fall will depend on how much money is really coming into the economy. Prices are @ $2.40 but it is entirely possible than inflows are well below the $17,280. Maybe $8,000, maybe $2,000. The price could be artificially supported by a combination of hoarding and manipulation.
If net inflows actually are <$8,000 then prices would be expected to fall (from $2.40) even after the reward is cut. They simply would fall slower than if the reward wasn't cut. On the other hand if net inflows are >$8K even if prices are supported today by manipulation and hoarding they would be supported by the fundamentals after the reward cut.
Net inflows is what matters. Obviously there are a lot of coins not being traded, sold, or exchanged right now. That is the hanging sword. If a significant number of them were exchanged for fiat then outflows would significantly increase and the net effect would be insufficient fund flows to support prices.
Difficulty changes don't drive price because regardless of difficulty mining is zero sum game. x coins are produced each day at difficulty of 20M or difficulty of 0.2M. Personally I think difficulty is going to fall 30%-40% even before the reward cut both in anticipation of the cut (as we get closer) and because more miners will realize prices aren't going to $30 and mining for a loss simply doesn't make sense.TL/DR version:
OF COURSE REWARD RATE HAS AN EFFECT ON PRICE. Lets say Satoshi had made the block reward 50 million coins per block instead of 50. Does anyone think prices would be ~$2.0 each ($16 trillion monetary base today)? Yes block reward change in the future is less of an effect (because it doesn't influence the coins already created) but it still is an effect.