1. [Native OS] I have version 1.2 installed. Is there any way to update without losing my saved progress in games?
I'm not sure about this, never updated native firmware because 1.2 is the latest version of official firmware, but I guess you can backup your save files before flashing, using a PC to copy all the saves to the hard drive, just in case...
2. [Native OS] How to change themes?
You need a custome firmware first: http://boards.dingoonity.org/dingoo-hacks/dingoonity-custom-firmware-1-2/
Search the firmware subforum for themes and detailed instructions.
3. [Native OS/Dingux] Is there any file manager (like the one in Native OS in "Browser"), but with ability to move and delete files?
For native I'm not sure, for Dingux there's DinguxCommander, but you can't access NAND from there, it can only modify the SDCard.
4. [Dingux] When creating a shortcut to game X in Dingux, what's the difference between "X.dge" and "X" without extension?
There's no difference, the extension of a file doesn't really mean anything under Linux based systems, some developers like to add the .dge while building it, some don't care. For example, is really common to have a file named "README" that's a text file, but it doesn't have a .txt extension.
Only make sure that you're adding a link to an executable file.
6. [Dingux] Why there are several games from this site : http://dl.openhandhelds.org/cgi-bin/dingoo.cgi, which after putting on SD card aren't working (they crash at startup, but there is A320 in the game's description)
You might be doing something wrong, those games should work. Is really hard to answer somethings as vague as that. What games? What did you tried? (some times it takes more than creating a link to make something work) What kernel do you have? Search the forum for any game that doesn't work and you'll find help.
7. [Native OS/Dingux] What is the best original Mario port?
Original Mario? A game with exactly the same levels, look and feel of the original? There's nothing like that, there are clones but I don't think they're like playing the original, just use a nes rom for that, it works without problems both on native and dingux emulators.
8. What system is more compatibile with games - Dingux or OpenDingux? Is Dingux still going to recieve future updates?
Dingux is more compatible at the moment because everything in the past was developed for it, and OpenDingux have a lot of things working but it's not even finished. It's like comparing compatibility of Windows XP and Vista after the first months of Vista release.
Anyway, a lot of emulators have been ported to OpenDingux and a lot of game engines and apps work too. And lately there were releases that only work in OpenDingux.
Legacy Dingux won't be updated, maybe some apps or games will, but I'm not sure if anyone is working for it.
9. [Native OS] How to check LCD number?
Please, do some research, that's been asked a million times and is really easy to find the answer with a quick google search, just type "dingoo lcd type check" or something like that. Look here: http://www.dingoo-scene.com/TheCompleteidiotsGuidetoDinguxWindoze.htm
In section "A - Flash firmware" it says: "Turn on your Dingoo. Go to the “System Setup” menu, then “About”, and press A. Then press the following key combination: up-right-down-up-right-down. Then you will see a hidden diagnostics screen."
11. How to check which version of Dingux I have installed and how to install, if possible, newer version?
Mmmh... legacy dingux kernels were made as separate projects, there's no real "newer" version (as in "based on an older version"), they just have different things. I think the best you can find is Science's kernel. I don't know how to check if you have it, maybe comparing kernel file size or creation dates with other kernels...