Thanks for the reply. My main concern is that my employer might be contacted, and subsequently start taking actions to replace me. Is this something I should be concerned about?
Phone interviews can be tough - best to do well away from company out of respect if nothing else, but as suggested a lunch break is a good time for this and a park with maybe an ear piece or Bluetooth headset is recommended. And it may seem obvious, but don't actually eat or drink at any point while on the call...
For in person, again just from a moral stand point, these should be conducted on your own time but emphasize your desire to get as many possible done on that day. Don't try to overbook though - the worst thing you can do is miss our all for a reschedule at the last minute. But you can do so if they have a healthy buffer between, especially if one is an earlier round (ie. Phone screen after and day of in person has been doable for me, but you definitely should refuel at that point and try to take a breather.
You absolutely should be aware of the market opportunities and the relative value and respect that can factor in. It is your right, and I would say or obligation, to do so and insure both you and your employer are benefiting equally. All that being said, your current company shouldn't be paying for this pursuit, so be fair and do it on your own time. In business, being consistently courteous and reasonable in all your pursuits will genuinely help your cause and best position for you to receive the same.
BTW, I love going to work. :-)