Some settings do not update unless you start playing on frame 0 or 1. This is just how Blender updates Simulation Nodes.

Unfortunately Simulation nodes operate on a per-frame basis so there aren't substeps to catch big positional changes. The best fix is to slow down your animation to half speed, do the sim, and then render every other frame (set frame step to 2)

This seems to be a 4.1 glitch. To fix add a new Geonodes modifier, and in its new node tree add the Set Material node. Now assign your material to that node.

This is a limitation of this system: it's more designed to create secondary motions without changing the base position too drastically. You can adjust the base mesh so they start in a more natural state and then increase the "Max Stretch Distance" setting to try to get more naturalistic effects.

Adjust your base mesh to be a bit tighter to the body from the start.

You may need to bake the geonodes sim. At the bottom of the Geonodes modifier you can set an appropriate file output for the cache (different for each object with the nodegroup). Then in the Physics tab you can bake the sim.

A good practice is to have 2 versions of those underneath objects: one that is used as the collider, and one that is used for rendering. That way you can mask out parts that should never be visible beneath clothing. Example: a character wearing pants and a shirt doesn't need their torso or legs visible. Create an instance of your base character, add a Vertex group selecting the torso and legs, add a Mask modifier and use that vertex group to make those parts unrenderable.

In addition to the body, you can use another cloth mesh as a collider.

Try increasing the Cloth Thickness: values between .02 and .07 are ideal.

Make sure you've properly set the Scale Correction input, or experiment with this value to see try to get a better result.

Add a Subdivision modifier before the Geo Fastcloth Geonodes modifier.