Well, in the deal of Windows 10: Every server forum I've checked, they all have a patch for Windows 10. I think is just one more .dll or I don't know since I use Windows 7.
About the "Private server realm":
I must say, depending if the server has any customizations, a Patch may contain these folders: System, Systextures, Animations,Textures, Staticmeshes, l2text
The red colored are the important ones, that are totaly necessary: system is the folder that allows you to connect to the server, systextures and animations are the ones that provide the customizations from the server (Not every server has customizations, so not every server have those folders in their patches)
I myself don't care to use the green ones, since Textures are just visual stuff, like changing Giran ground with grass and stuff. I don't even know what Staticmeshes does, and l2text is useless.
To "Patch" your client:
1- Open the folders you downloaded, copy and paste them into your Lineage 2 directory. (Systextures downloaded to systextures of your installed client).
2- Once you did the same with all the folders of the patch, Open "system" folder, Run as Administrator L2.exe (To avoid problems)
About the client, it all depends on what Chronicle you want to play, the most popular ones are Interlude (C6) and High Five (H5/CT2.6) so you'll have to download the Client of the chronicle of your server.
Interlude is more classic while High Five is very interesting but more complicated. You should try both and decide which to play. (I'm an Interlude player, but I still have the H5 for some cases)
I haven't seen an Ertheia private server, but there might be some.
If you have any other doubts don't worry to answer to this topic or Pm me.
