Can I (should I) do anything about these open PVC exhaust pipes in regards to rain or pests
Your setup looks completely standard to me. In my region we have converted most HVAC systems and water heaters to high efficiency units. They vent nearly room-temperature gas via PVC in most cases. The inlets and outlets are arranged to prevent recirculation of exhaust gases into your home, and the outlets have mechanisms inside to deal with drainage from condensation and a bit of rain.
Regarding rodents... it's possible. About once a year I get mice in my suburban home, and I suspect the HVAC plumbing as an ingress point. However, it can be dangerous to apply mesh. Accumulation of dust, lint, insect nests, natural material, or whatever can obstruct intake and exhaust flow, potentially causing health and safety issues. I'd avoid doing so.
My $.03? Don't change anything unless there's an actual problem. If you must, consult an HVAC specialist for the right way to deal with the specific issue. Mesh covers may be available that provide a large surface area unlikely to clog, for example.