I recommend reading this blog post first. It’s a walk through on how to create your own reciptes in your own layer. Unfortunately it’s on a now outdated vesion of openembedded though.
How do I add this service to my oe-core build so that the subsequent build will have this service by default?
You could create a patch which changes the original autostart file and use a bbappend
lxde-common_%.bbappend from your layer to apply this patch.
Another question is How do I add my scripts to the Rootfs in the oe-core build?
To only installing files into the rootfs take this recipe as an example. Then add the package created to IMAGE_INSTALL in your image recipe.