Juuso,
1. No, you do not need to load 4.2.2 in order to load CM 10.1.3 - CM is a complete replacement for the OS version currently on your phone (and is equivalent to Android version 4.2.2).
2. The GAPPS package that you need to load is the one dated 20130812 - that is the appropriate version for Android 4.2.2 (as shown in the table at the top of the page at goo.im/gapps that you pointed to).
3. You can use GooManager:
https://forums.androidcentral.com/e...details?id=com.s0up.goomanager&token=A-i4whxb to help with keeping your ROM updated.
Sent from my rooted, debloated, deodexed Sinclair ZX-80 running CM 10.2 using Tapatalk 2