What are the purposes of the struts and lateral bracing?
First I'm not a structural engineer but I do understand laterial bracing. Lateral bracing is provided on compression members (which are like columns) to reduce the possibility of buckling. In the case beams or elements subjected to bending the upper flanges are in compression and it is standard practice to provide laterial bracing. Not sure what is meant by struts. Unless your talking about cross braces.
What exactly is a landing gear side load?
JetDoc pretty much has it. When you land in a crosswind, your airplane and your wheels are pointed in a slightly different direction than your direction of travel. As your wheels contact theground, the wheels want to go in the direction they are pointed, and the inertia of the airplane keeps it going in the same direction it was travelling in. The difference in the direction of travel, and the direction the wheels are pointed, causes the tires and wheels and landing gear struts to be pulled sideways. If the difference is extreme, the tires can pull away from the rims. Landing gear struts are designed to take a straight on load as you land, and the side force could make the strut fail, or pull away from its mount.