Second question first. Depends on what level government you are talking about. Federal government relaxed its rules a few years ago to let government travelers keep their miles for government trips. So, you should check if your governmental unit allows it.

As to most cost savings, you'll have to check the majors against the low fare airlines. Make sure to check Southwest.

However, don't disregard the possible savings coming from combining your government miles with your private miles. For example, if you travel for government primarily on Delta, and combining these 6 round trips with your government miles would give you an extra ticket (or boost you up to another level, like a biz class ticket to Europe instead of an econ. class), then it might make sense to fly the airline which would combine the best with your government business travel miles.