Think about how your drive train is designed - the whole idea is to have a chain that can move sideways from one sprocket to another.
Sometimes a bad rattle is enough to upset the chain sufficiently to jump to another sprocket. A light nudge sideways on the chain or the derailer can do it. Unless the chain falls off completely, don't worry about it. If it does, check limit screws and chain length. Derailing is annoying but usually not harmful. Jamming the chain in between spokes and sprocket can damage the spokes though. Chain suck - when the chain fails to disengage the chainring and bunch up instead can damage both front derailer and frame, depending on where it happens.

The noise is probably a squeak brought on by you putting more effort into the bike. These can be real hard to track down. Can be in the bar/stem/fork interfaces, can be bottom bracket, can be saddle, can be pedals... disassemble, lube and reassemble are the usual recommendations.

I don't think you really see the spokes bending. That was the case, the wheel would collapse. And it looks like 32-spoke, which should be sturdy enough for most use.
Stationary, regular wheels are quoted to stand something like 500-600 lbs before collapsing. Dynamically, it'll be less of course. But outside a crash scenario, the likely killer of a wheel is either spoke fatigue, brake track wear or point damage to the rim from hitting a curb or a pothole.