Whatever you do, do not use copper grease on your caliper pistons, as it will affect the seals and cause failure.
Use a silicon based rubber grease.
How old is the bike? Are they the original brake lines?
It varies from manufacturer to manufacturer, but if your lines are more than 5 years old and not braided steel, you really should change them.
Other than the curious effect you have described with the clamping, I would have suggested raising the calipers above the master cylinder when you bleed them. You'd be suprised how tricky air can be to get rid of, sometimes.
