Backup your Aircraft.cfg. Scroll down to the [contact_points] section and replace the first three entries with these.
point.0=1, -18.00, 0.00, -6.55, 3600, 0, 1.0350, 75.0, 0.20, 4.20, 0.610, 3.0, 3.0, 0, 0.0, 0.0
point.1=1, -35.50, -6.00, -7.60, 3600, 1, 1.2349, 0.0, 1.95, 1.00, 0.500, 3.9, 3.9, 2, 0.0, 0.0
point.2=1, -35.50, 6.00, -7.60, 3600, 2, 1.2349, 0.0, 1.95, 1.00, 0.500, 3.7, 3.7, 3, 0.0, 0.0
They address the height issue and make your aircraft look more visually realistic. But also affect the bouncing and Damping of the gears (nose will sink in on carrier arrests). There are more coordinates if you search around. But these "look" better than others.
As for the bouncing thing, it's probably a scenery issue. Does it happen all the time and everywhere (runway, apron, grass, etc.)?