[Galaxy S3 Neo Original Android Development] Humaxxx:
Deutsch wäre einfacher ;)

MY answers about:

Deep sleep: Its hard to descripe what deep sleep really is / does. But in easy words its a battery safer feature, that shuts down a lot of processes to safe battery, but your are still able to get notifications if you get a mail / message or something else. You can do it still the way you did on Stock ROM. After charging just do a reboot and deep sleep is again working until you plug in a charger again.

Overlay buttons: I think you mean software-buttons. These are disabled, but you can enabled it and disable the hardware-buttons if you want to. For me i like to stay with hardware - buttons...

Compatibility considerations: Yes some apps wont longer work with older Android-Versions. With Custom Rom you have the choice of CM11 (Android 4.4 - dont remember if it was official or unofficial), CM12.1 (Android 5.1.1 - official nightly), LOS13 (Android 6 unofficial) , LOS14.1 (Android 7.1.2 unofficial), LOS15.1(Android 8 unofficial), LOS16 (Android 9 official), LOS17.1 (Android Q (10) - at the moment unofficial and under development). Also there are other Roms like Noble Rom (Themed Android 5.1.1 or 6 - i am not sure) here on XDA, Resurection Remix (Android 5.1.1) and many custom Roms on based on Android 7.1.2 like AOKP. All Roms habe other small issues / bugs. So it can depend on your preferencies with rom to flash... For ME good roms are CM12.1, LOS14.1, LOS16 and AOKP. ALso LOS17.1, but at the moment is beta and no root possible.

Security considerations: I guess the Custom Roms are at least as safe as the stock rom, but with updated security patches! Refering to LOS 16 a dev told me, that the security patch date of the blobs are latest extracted from stock rom, there are no newer available because Samsung doesnt support S3Neo anymore. But other security Patches are up to date with LOS16

functionality (mainly security) and usability issues: i dont remember if i was using to try xposed on los14 or 16. But it should work! Greenify should also work, the others i havent any experience... But i recommend to first set up LOS16, then make a backup in twrp (Boot/System/Data). And then try these apps. If anything goes wrong you can restore your backup in 5 minutes... Also i recommend to test first the LOS16 security features, maybe they are enough for you.

block all calls except a list of important contacts when I'm busy or sleeping: LOS16 supports "Do not disturb mode" - which exactly does what you want!

incoming call mute gesture - I dont use any gestures. But LOS 16 supports it, but i think in an other way instead of turning of the phone to the other side you can press Power-button and louder-button to either let virbrate or mute the phone (although turning the phone over is probably easier). But you must try by yourself browsing through the LOS settings...

LED status: I dont know exactly what you mean. But in LOS16 you can set up for every App a color for LED-Notification, also at what speed the LED should flash

The flashing process: based on your used stock rom (I9301IXXUAQL1_I9301ITTRAQL1_I9301IXXUAPG1) i guess you have GT-I9301I. Depending on your stock firmware you should now have the latest baseband, but not the latest bootloader. It can work, but its not sure. I recommend to flash the refered greek stock rom to be sure - and you dont have to matter if this stock rom has a branding, because you are flashing a new rom after that. Also on S3Neo you can completly undo all changes by flashing your original stock rom! Before you begin to flash you should check some things! Download app "HW DEVICE INFO" and grant root permission. Then under "system" note your baseband, bootloader and under "camera" note the "code" of your camera because S3Neo has either Samsung Cam or Sony Cam. Sometime "Camera Check" is not correct, so you have to try which cam you have by flashing a rom - if cam is not working you know you have to choose the other one. So i recommend before setting up anything test if camera is working, otherwise you have to set up everything again... Latest baseband ends with "...uapg1", latest bootloader with "xcuara1". After flashing the greek stock rom boot up and check with the app device info if both has updated, because sometimes in needs more than 1 or 2 flash to update baseband / bootloader. -- AHH ok, you already checked that you have Sony-Cam :)

Here the steps for flashing los16

0) Copy neccesary files to your sd card (LOS16 ,Opengapps, Magisk 20.3)

1) Check if baseband / bootloader is latest. if not flash greek stock rom via odin until it has updated

2) flash twrp (it must be 3.3.1-1) via odin. No need for adb. Make sure you untick "auto-reboot" in odin.

3) Boot into twrp. Do a backup of "EFS" i dont know if EFS is available if you are on stock rom, if not do a full backup! EFS is important because while flashing there can be something go wrong and EFS stores information about your IMEI and Bluetooth infos...

4) After Backup has completed do a full wipe, wiping everything except "SD-Card" and "USB-OTG"

5) Flash LOS

6) Flash Gapps if you want

7) Flash Magisk or LOS SU if you want

8) reboot - first flash can last some time...