This isn\'t about the different methods I could or should be using to utilize the queues in the best manner, rather something I have seen happening that makes no sense to me.
question 1: If you're using a synchronized queue, then: no, you're safe! But you'll need to use the synchronized instance on both sides, the supplier and the feeder.
question 2: Terminating your worker thread when there is no work to do, is a simple job. However, you either way need a monitoring thread or have the queue start a background worker thread whenever the queue has something to do. The last one sounds more like the ActiveObject Pattern, than a simple queue (which's Single-Responsibily-Pattern says that it should only do queueing).
In addition, I'd go for a blocking queue instead of your code above. The way your code works requires CPU processing power even if there is no work to do. A blocking queue lets your worker thread sleep whenever there is nothing to do. You can have multiple sleeping threads running without using CPU processing power.
C# doesn't come with a blocking queue implementation, but there a many out there. See this example and this one.