...unless you're in a jet with fuselage-mounted engines (less asymmetrical thrust), and you're lightly loaded. In that case, wait as long as possible to start up the second one to keep from sliding when you frequently have to apply the brakes during taxi.
Yes, but this could also get you in trouble. Remember that when the engine isn't running, it could be building up contaminants that will just get sucked in once it starts. Even if the cowling appears clear, it could be buidling ice on the blades or spinner. On the 737 I believe that we're supposed to keep them both running after we've been deiced.