The blockchain won't "halt" at overflow, it will have validation problems. The overflow doesn't automatically stop the chain. Nodes would continue with wrapped-around timestamps (though this would cause some problems).
Yes the blockchain halts.
The timestamps are required to eventually increase; more specifically each new timestamp is required to be strictly greater than the MTP timestamp. Assuming these timestamps stay reasonably accurate up to the end then at best you can squeeze out a few more blocks by lying about the time, but eventually the MTP timestamp will reach its maximum value. At that point there are no more valid blocks that can be appended to the chain.