I'll try and explain what is going on without getting too technical. ClearNight disables and replaces the rust environment syncing between the server and each individual client. Due to this, each client will technically be out of sync with the server allowing the server to progress time naturally while allowing for a full moon on the client's end all the time. However, due to the server and clients being out of sync we start experiencing some odd behaviours like you're experiencing.
For example, if the servers time is in the summer months you will have long days with short nights, but if the current full moon date is in the winter months you will have short days with longer nights. This is what causes the moon to rise then jump into position, the server doesn't think the night has started yet, but it has started on the client. Once night starts on the server, an update gets sent to the client causing it to jump.
Now for the issues with weather, that is an entirely new beast that I am still trying to tackle. ClearNight has a list of full moon dates that are rotated at the start of a new day which is 00:00. Each of these dates has a weather pattern associated with them now, so each time a new day rolls around the date will switch along with the weather associated with that day.
Hopefully that helps explain what is going on. I have a big update in the works, shoot me a PM if you want to test it out.