The reward is just a special transaction called a coinbase and will get dropped just like any other transaction that is part of a stale block. 
It kind of depends on the softfork itself and the role of the node. If for example the block size is lowered and the node is a miner it can mine blocks that are larger than the consensus allows. This can be a problem for the miner since his blocks will not get accepted if the majority of nodes updated to the newer version. In a segwit fork this is not the case since the block size remained the same and both miners wold be able to continue mining valid blocks.
Well if a miner continued to mine on the old chain and eventually updated their nodes, all their blocks would become invalid. He would loose all the work he has done, so better update fast 
Not neccessarily, Monero for example has hard forks all the time. It is its method of staying ASIC resistant. 