With casper stage 1 being introduced into Ethereum, the chain will be shaped by an hybrid POW/POS consensus mechanism. In this stage, every 100th block is going to be finalized by the POS mechanism and will be regarded as a checkpoint (https://github.com/ethereum/research/wiki/Casper-Version-1-Implementation-Guide). The idea behind checkpoint blocks is, that any miner following the rules in this release of ethereum has to accept this block, no choice given. All other 99 blocks are going to be mined as usual. To answer your second question, mined blocks and POS blocks will both be used on the same chain.
So why should miners continue? If miners leave, the difficulty to hash a new block will decrease and it will be more lucrative for other miners to join in, what they will do for sure at a certain point. Ethereum will go on with its path. If POS can be used as the only consensus protocol and is as effective as POW, it will be used for sure. To name only two advantages, it does consume less power (no hashing required where almost all hashes are useless) and the barrier for new people willing to participate in the decision for the next block is easier to overcome. It is possible that POS might become the only consensus mechanism. Until that happens, I would try to mine as much ether as possible if I would be a miner. By doing so I potentially increase my reward from the POS system by being able to place a much higher stake.